Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Splitting theorems and geometric aspects of
overdetermined problems
Alberto FARINA
Université de Picardie J. Verne
LAMFA, CNRS UMR 7352
Amiens, France
CIMPA school
Santiago (Cile), April 8-15, 2015
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Introduction and aims
Aims
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Introduction and aims
Aims
We propose a unified approach to three different topics in a Riemannian
setting:
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Introduction and aims
Aims
We propose a unified approach to three different topics in a Riemannian
setting:
splitting theorems,
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Introduction and aims
Aims
We propose a unified approach to three different topics in a Riemannian
setting:
splitting theorems,
symmetry results,
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Introduction and aims
Aims
We propose a unified approach to three different topics in a Riemannian
setting:
splitting theorems,
symmetry results,
overdetermined elliptic problems.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Introduction and aims
Roughly speaking, by the existence of a suitable solution to a
semilinear PDE on a Riemannian manifold (submanifold) we are able
to classify both the solution and the manifold (submanifold).
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Introduction and aims
Roughly speaking, by the existence of a suitable solution to a
semilinear PDE on a Riemannian manifold (submanifold) we are able
to classify both the solution and the manifold (submanifold).
The approach provides a nonlinear (PDEs) criterium for the study of
the above mentioned (apparently unrelated) topics and, to the best
of our knowledge, is new.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Introduction and aims
Roughly speaking, by the existence of a suitable solution to a
semilinear PDE on a Riemannian manifold (submanifold) we are able
to classify both the solution and the manifold (submanifold).
The approach provides a nonlinear (PDEs) criterium for the study of
the above mentioned (apparently unrelated) topics and, to the best
of our knowledge, is new.
We also hope that it will be useful to better understand the situation
even in the special case of the Euclidean space RN , which was the
starting point of our investigations.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
A model in fluid mechanics (R. L. Fosdick)
The problem
(
−∆u = 1
∂u
u = 0, ∂ν
= const.
in Ω
on ∂Ω
is used to study a viscous incompressible fluid moving in straight parallel
streamlines through a straight pipe with planar cross section Ω ⊂ R2 .
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
A model in fluid mechanics (R. L. Fosdick)
The problem
(
−∆u = 1
∂u
u = 0, ∂ν
= const.
in Ω
on ∂Ω
is used to study a viscous incompressible fluid moving in straight parallel
streamlines through a straight pipe with planar cross section Ω ⊂ R2 .
In this setting
u represents the flow velocity of the fluid,
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
A model in fluid mechanics (R. L. Fosdick)
The problem
(
−∆u = 1
∂u
u = 0, ∂ν
= const.
in Ω
on ∂Ω
is used to study a viscous incompressible fluid moving in straight parallel
streamlines through a straight pipe with planar cross section Ω ⊂ R2 .
In this setting
u represents the flow velocity of the fluid,
the Dirichlet condition is the adherence condition to the wall,
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
A model in fluid mechanics (R. L. Fosdick)
The problem
(
−∆u = 1
∂u
u = 0, ∂ν
= const.
in Ω
on ∂Ω
is used to study a viscous incompressible fluid moving in straight parallel
streamlines through a straight pipe with planar cross section Ω ⊂ R2 .
In this setting
u represents the flow velocity of the fluid,
the Dirichlet condition is the adherence condition to the wall,
∂u
∂ν
is the tangential stress per unit area on the pipe wall.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Question (R. L. Fosdick, 1971)
Assuming that the adherence condition holds on the entire pipe wall, and
that the tangential stress is constant everywhere on the wall, is it true
that the pipe has a circular cross section ?
(i.e. Is true that Ω a disk ?)
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Serrin’s problem
To answer Fosdick’s question, J. Serrin considered the general
overdetermined boundary value problem


in Ω
−∆u = f (u)
u>0
in Ω


∂u
= const. on ∂Ω
u = 0, ∂ν
(1)
where f is a smooth function and Ω ⊂ RN , N ≥ 2, smooth and bounded.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Serrin’s problem
and proved the following celebrated result
Theorem 1 (J. Serrin, 1971)
Let Ω ⊂ RN be a connected bounded open set of class C 2 and let f ∈ C 1 .
If the overdetermined boundary value problem (1) admits a C 2 (Ω), then
Ω must be a ball and u is radially symmetric about its center.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Serrin’s problem
and proved the following celebrated result
Theorem 1 (J. Serrin, 1971)
Let Ω ⊂ RN be a connected bounded open set of class C 2 and let f ∈ C 1 .
If the overdetermined boundary value problem (1) admits a C 2 (Ω), then
Ω must be a ball and u is radially symmetric about its center.
Serrin introduced the PDE community to the celebrated moving
planes method (based on Alexandrov’s reflexion principle).
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Serrin’s problem
and proved the following celebrated result
Theorem 1 (J. Serrin, 1971)
Let Ω ⊂ RN be a connected bounded open set of class C 2 and let f ∈ C 1 .
If the overdetermined boundary value problem (1) admits a C 2 (Ω), then
Ω must be a ball and u is radially symmetric about its center.
Serrin introduced the PDE community to the celebrated moving
planes method (based on Alexandrov’s reflexion principle).
A huge amount of (important) results and researches originated
from the seminal work of J. Serrin (Gidas-Ni-Nirenberg).
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Serrin’s problem in unbounded domains
In 1997, H. Berestycki, L. Caffarelli and L. Nirenberg, motivated by
questions on the regularity of some one-phase free boundary problems,
are led to the study of semilinear problems of bistable type in globally
Lipschitz unbounded domains.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Serrin’s problem in unbounded domains
In 1997, H. Berestycki, L. Caffarelli and L. Nirenberg, motivated by
questions on the regularity of some one-phase free boundary problems,
are led to the study of semilinear problems of bistable type in globally
Lipschitz unbounded domains.
Those kind of domains appear as ”limiting domains” after a standard
blow-up procedure ”near” the free boundary in the original problem.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
A typical ex. of bistable nonlinearity is given by
f (u) = u(1 − u 2 )
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
A typical ex. of bistable nonlinearity is given by
f (u) = u(1 − u 2 )
⇓
−∆u = u − u 3
the Allen-Cahn equation.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Conjecture (Berestycki, Caffarelli, Nirenberg, 1997)
Assuming that Ω ⊂ RN , N ≥ 2, is a smooth domain with Ωc connected
and that there exists a bounded smooth solution of the overdetermined
boundary value problem


in Ω
−∆u = f (u)
(2)
u>0
in Ω


∂u
u = 0, ∂ν = const. on ∂Ω
for some Lipschitz function f, then Ω is either a half-space, a ball, a
circular-cylinder-type domain: Rj × B, with B a ball in RN−j or the
complement of one these regions.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Extremal domains for the Dirichlet-Laplace operator
Definition
Let Ω be a smooth bounded domain of a smooth, complete Riemannian
manifold (M, h , i).
Let λΩ be the first eigenvalue of −∆ in Ω with homogeneous Dirichlet
boundary conditions.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Extremal domains for the Dirichlet-Laplace operator
Definition
Let Ω be a smooth bounded domain of a smooth, complete Riemannian
manifold (M, h , i).
Let λΩ be the first eigenvalue of −∆ in Ω with homogeneous Dirichlet
boundary conditions.
The set Ω is an extremal domain if it is a critical point of the shape
functional
Ω −→ λΩ
with respect to smooth variations of the domain which preserve its
volume.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Extremal domains are characterized as the domains for which one can
solve the following overdetermined bvp :

−∆u = λΩ u
in Ω ,



u > 0
in Ω,

u=0
on ∂Ω,



h∇u, νi = const. on ∂Ω,
(3)
where ν is the outward unit normal vector to ∂Ω.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Extremal domains for the Dirichlet-Laplace operator
In the Euclidean case (RN , h , ican ), the only extremal domains are balls,
thanks to the result of J. Serrin.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Extremal domains for the Dirichlet-Laplace operator
In the Euclidean case (RN , h , ican ), the only extremal domains are balls,
thanks to the result of J. Serrin.
It is natural to ask :
Find ”reasonable” conditions on the Riemannian manifold (M, h , i) and
on the open domain Ω leading to a classification of Ω and/or u.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
A conjecture of De Giorgi
In 1978 E. De Giorgi formulated the following question :
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
A conjecture of De Giorgi
In 1978 E. De Giorgi formulated the following question :
Let u ∈ C 2 (RN , [−1, 1]) satisfy
− ∆u = u − u 3
and
A. FARINA
∂u
>0
∂xN
on RN .
(4)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
A conjecture of De Giorgi
In 1978 E. De Giorgi formulated the following question :
Let u ∈ C 2 (RN , [−1, 1]) satisfy
− ∆u = u − u 3
and
∂u
>0
∂xN
on RN .
(4)
Is it true that all the level sets of u are hyperplanes, at least if N ≤ 8?
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
A conjecture of De Giorgi
It is easily seen that the claim of the conjecture is equivalent to :
there exist a unit vector a = (a1 , ..., aN ) ∈ RN , with aN > 0, and a
function g ∈ C 2 (R) such that :
u(x) = g (a · x)
A. FARINA
∀ x ∈ RN .
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
A conjecture of De Giorgi
It is easily seen that the claim of the conjecture is equivalent to :
there exist a unit vector a = (a1 , ..., aN ) ∈ RN , with aN > 0, and a
function g ∈ C 2 (R) such that :
u(x) = g (a · x)
∀ x ∈ RN .
By an ODE analysis we have that, for some α ∈ R :
s + α
g (s) = tanh √
∀ s ∈ R.
2
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
”Physical derivation of the model”
We are given some substance in a container Ω, which may exhibit two
phases, which we label with “−1” and “+1”, and
we would like to describe mathematically the pattern and the
separation of such phases.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
The classical Van der Waals/Cahn-Hilliard theory tell us that interface
formation is driven by a variational principle, that is the pattern is the
outcome of the minimization of an energy of the form :
Z
Eo (u; Ω) =
W (u(x)) dx
Ω
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
The classical Van der Waals/Cahn-Hilliard theory tell us that interface
formation is driven by a variational principle, that is the pattern is the
outcome of the minimization of an energy of the form :
Z
Eo (u; Ω) =
W (u(x)) dx
Ω
u(x) represents the state of the substance at the point x ∈ Ω.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
The classical Van der Waals/Cahn-Hilliard theory tell us that interface
formation is driven by a variational principle, that is the pattern is the
outcome of the minimization of an energy of the form :
Z
Eo (u; Ω) =
W (u(x)) dx
Ω
u(x) represents the state of the substance at the point x ∈ Ω.
W is some “double well” function, i.e., such that W (±1) = 0 and
W (r ) > 0 if r 6= ±1
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
The classical Van der Waals/Cahn-Hilliard theory tell us that interface
formation is driven by a variational principle, that is the pattern is the
outcome of the minimization of an energy of the form :
Z
Eo (u; Ω) =
W (u(x)) dx
Ω
u(x) represents the state of the substance at the point x ∈ Ω.
W is some “double well” function, i.e., such that W (±1) = 0 and
W (r ) > 0 if r 6= ±1
The typical ex. is
W (u) =
(u 2 −1)2
4
A. FARINA
=⇒
Allen-Cahn model.
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
The model is not a satisfactory : any function attaining only the values
−1 and +1 minimizes the energy Eo .
In particular, the separation between the two phases could be as wild as
possible and the energy would not be affected!
Since this is not the case in physical applications one usually adds to Eo ,
a term that penalizes the formation of unnecessary interfaces.
This is accomplished by adding to Eo a small gradient term, that is by
looking at the energy
Z
1
E (u; Ω) =
|∇u(x)|2 + W (u(x)) dx,
2
Ω
where > 0 is a small parameter.
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Splitting theorems and geometric aspects of overdetermined problems
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A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Such small gradient term indeed cuts the interfaces as much as possible,
in the sense that the minimizers of E turn out to be smooth functions,
taking values between −1 and +1, and whose level sets approach (in a
suitable way) hypersurfaces of least possible area, when → 0+
[Modica-Mortola (1977), Modica (1987), Caffarelli-Cordoba (1995)].
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Splitting theorems and geometric aspects of overdetermined problems
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A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Possible motivation for the conjecture
Let u be as in De Giorgi’s conjecture, > 0 and let
u (x) := u(x/)
The monotonicity assumption in De Giorgi’s conjecture seems to suggest
that :
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Splitting theorems and geometric aspects of overdetermined problems
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A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Possible motivation for the conjecture
Let u be as in De Giorgi’s conjecture, > 0 and let
u (x) := u(x/)
The monotonicity assumption in De Giorgi’s conjecture seems to suggest
that :
the level sets of u (and thus those of u ) are graphs over RN−1 .
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Possible motivation for the conjecture
Let u be as in De Giorgi’s conjecture, > 0 and let
u (x) := u(x/)
The monotonicity assumption in De Giorgi’s conjecture seems to suggest
that :
the level sets of u (and thus those of u ) are graphs over RN−1 .
the phase transition happens in a straight, minimal way.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Thus, when → 0+ , the level sets of u are closer and closer to global
minimal graphs ϕ over RN−1 , i.e. a solution of
"
∇ϕ
−div p
1 + |∇ϕ |2
A. FARINA
#
= 0 in RN−1
Splitting theorems and geometric aspects of overdetermined problems
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A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Thus, when → 0+ , the level sets of u are closer and closer to global
minimal graphs ϕ over RN−1 , i.e. a solution of
"
∇ϕ
−div p
1 + |∇ϕ |2
#
= 0 in RN−1
Since global minimal graphs are flat for N − 1 ≤ 7, due to Bernstein-type
Theorems, it follows that the level sets of u are close to an hyperplane.
Here
N ≤ 8 is crucial !
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Splitting theorems and geometric aspects of overdetermined problems
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A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
Now, since elliptic problems are somehow rigid, we may suspect that once
{u = c} is close enough to a hyperplane, it is a hyperplane itself.
By scaling back, this would give that {u = c} is a hyperplane.
Then, the level sets of u would be parallel hyperplanes as asked by De
Giorgi’s conjecture.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
In the previous (heuristic) argument some gaps have to be filled :
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
In the previous (heuristic) argument some gaps have to be filled :
no minimality condition is explicitly required in De Giorgi’s
conjecture, so the results about the asymptotic behavior of
minimizers are not directly applicable.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
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The framework
Splitting theorems
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Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
In the previous (heuristic) argument some gaps have to be filled :
no minimality condition is explicitly required in De Giorgi’s
conjecture, so the results about the asymptotic behavior of
minimizers are not directly applicable.
the monotonicity condition does not assure, in principle, that the
level sets of u are entire graphs over RN−1 , so Bernstein-type results
are not directly applicable,
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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Overdetermined boundary value problems
References
A model in fluid mechanics
Serrin’s problem
Serrin’s problem in unbounded domains
BCN conjecture
Geometry and shape optimization : extremal domains
A problem arising in phase transitions and a conjecture of De Giorgi
”Physical derivation of the model”
Possible motivation for the conjecture
In the previous (heuristic) argument some gaps have to be filled :
no minimality condition is explicitly required in De Giorgi’s
conjecture, so the results about the asymptotic behavior of
minimizers are not directly applicable.
the monotonicity condition does not assure, in principle, that the
level sets of u are entire graphs over RN−1 , so Bernstein-type results
are not directly applicable,
one would need to prove the rigidity argument.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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Splitting theorems
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The framework
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Examples of stable solutions
Our point of view :
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Splitting theorems and geometric aspects of overdetermined problems
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The framework
Splitting theorems
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The framework
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Examples of stable solutions
Our point of view :
We embed overdetermined bvp(s) and symmetry problems, into the
general framework of splitting theorems.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The framework
Stable solutions
Examples of stable solutions
Our point of view :
We embed overdetermined bvp(s) and symmetry problems, into the
general framework of splitting theorems.
We interpretate those topics as two different aspects or expressions
of a unique phenomenon : a splitting theorem.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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The framework
Stable solutions
Examples of stable solutions
(M, h , i) is a smooth complete, Riemannian manifold without boundary,
with Ric ≥ 0,
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Splitting theorems and geometric aspects of overdetermined problems
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The framework
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Examples of stable solutions
(M, h , i) is a smooth complete, Riemannian manifold without boundary,
with Ric ≥ 0,
which posses a solution of
(
−∆u = f (u)
u 6= const.
on M
(5)
with f ∈ C 1 (M)
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Splitting theorems and geometric aspects of overdetermined problems
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The framework
Stable solutions
Examples of stable solutions
(M, h , i) is a smooth complete, Riemannian manifold without boundary,
with Ric ≥ 0,
which posses a solution of
(
−∆u = f (u)
u 6= const.
on M
(5)
with f ∈ C 1 (M)
u is a stable solution of (5).
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Splitting theorems and geometric aspects of overdetermined problems
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The framework
Stable solutions
Examples of stable solutions
Solutions of (5) are critical points of the energy functional E given by
1
E(w ) =
2
Z
2
|∇w | dx −
Z
Z
F (w )dx,
where
F (t) =
t
f (s)ds, (6)
0
with respect to compactly supported variations.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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The framework
Stable solutions
Examples of stable solutions
Stable solutions
Definition
The function u is said to be a stable solution of (5) if
Z
Z
2
|∇φ| dx −
f 0 (u)φ2 dx ≥ 0
for every φ ∈ Cc∞ (M)
M
(7)
M
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Splitting theorems and geometric aspects of overdetermined problems
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The framework
Stable solutions
Examples of stable solutions
Stable solutions
Definition
The function u is said to be a stable solution of (5) if
Z
Z
2
|∇φ| dx −
f 0 (u)φ2 dx ≥ 0
for every φ ∈ Cc∞ (M)
M
(7)
M
or,
if the the Jacobi operator of E at u,
Jφ = −∆φ − f 0 (u)φ
∀ φ ∈ Cc∞ (M),
(8)
is non-negative on Cc∞ (M).
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Splitting theorems and geometric aspects of overdetermined problems
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The framework
Stable solutions
Examples of stable solutions
Stable solutions
Typical examples of stable solutions to (5) are :
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Splitting theorems and geometric aspects of overdetermined problems
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Splitting theorems
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The framework
Stable solutions
Examples of stable solutions
Stable solutions
Typical examples of stable solutions to (5) are :
local minimizers of the energy functional E, given by (6).
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
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The framework
Stable solutions
Examples of stable solutions
Stable solutions
Typical examples of stable solutions to (5) are :
local minimizers of the energy functional E, given by (6).
( harmonic functions ),
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Splitting theorems and geometric aspects of overdetermined problems
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Motivations and examples
The framework
Splitting theorems
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The framework
Stable solutions
Examples of stable solutions
Stable solutions
Typical examples of stable solutions to (5) are :
local minimizers of the energy functional E, given by (6).
( harmonic functions ),
”monotone ” solutions.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Splitting theorem
Theorem
Let (M, h , i) be a complete, Riemannian manifold without boundary,
satisfying Ric ≥ 0. Suppose that u ∈ C 3 (M) be a non-constant, stable
solution of −∆u = f (u), for f ∈ C 1 (R). If either
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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The framework
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Splitting theorem
Theorem
Let (M, h , i) be a complete, Riemannian manifold without boundary,
satisfying Ric ≥ 0. Suppose that u ∈ C 3 (M) be a non-constant, stable
solution of −∆u = f (u), for f ∈ C 1 (R). If either
(i) M is parabolic and ∇u ∈ L∞ (M), or
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Splitting theorem
Theorem
Let (M, h , i) be a complete, Riemannian manifold without boundary,
satisfying Ric ≥ 0. Suppose that u ∈ C 3 (M) be a non-constant, stable
solution of −∆u = f (u), for f ∈ C 1 (R). If either
(i) M is parabolic and ∇u ∈ L∞ (M), or
(ii) The function |∇u| satisfies
Z
|∇u|2 dx = o(R 2 log R)
as R → +∞.
(9)
BR
Then,
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Splitting theorems and geometric aspects of overdetermined problems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Splitting theorem
Theorem
M = N × R with the product metric h , i = h , iN + dt 2 , for some
complete, totally geodesic, parabolic hypersurface N with RicN ≥ 0.
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Splitting theorems and geometric aspects of overdetermined problems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Splitting theorem
Theorem
M = N × R with the product metric h , i = h , iN + dt 2 , for some
complete, totally geodesic, parabolic hypersurface N with RicN ≥ 0.
In particular,
M = R2 or S1 × R, with their flat metric, if m = 2;
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Splitting theorems and geometric aspects of overdetermined problems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Splitting theorem
Theorem
M = N × R with the product metric h , i = h , iN + dt 2 , for some
complete, totally geodesic, parabolic hypersurface N with RicN ≥ 0.
In particular,
M = R2 or S1 × R, with their flat metric, if m = 2;
u depends only on t, has no critical points, and writing u = y (t) it
holds that −y 00 = f (y ).
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Splitting theorems and geometric aspects of overdetermined problems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Compact setting
Corollary
Let (M, h , i) be a compact Riemannian manifold without boundary,
satisfying Ric ≥ 0. Suppose that u ∈ C 3 (M) is a stable solution of
−∆u = f (u), for some f ∈ C 1 (R).
Then, u must be constant.
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Splitting theorems and geometric aspects of overdetermined problems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
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Parabolicity
Definition
M is parabolic if any non-negative superharmonic function on M is
constant,
that is
(
−∆w ≥ 0
w ≥0
on M
on M
A. FARINA
=⇒
w ≡ const.
(10)
Splitting theorems and geometric aspects of overdetermined problems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Parabolicity
Theorem
A Riemannian manifold M is parabolic if and only if
∃ a sequence {φn }n≥1 , φn ∈ Cc0,1 (M), 0 ≤ φn ≤ 1 and
Z
{φn } is monotone increasing to 1,
lim
|∇φn |2 = 0.
n→+∞
A. FARINA
(11)
M
Splitting theorems and geometric aspects of overdetermined problems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒
In the compact case, just take φn ≡ 1 for every n ≥ 1.
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Splitting theorems and geometric aspects of overdetermined problems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒
In the compact case, just take φn ≡ 1 for every n ≥ 1.
In the non-compact case we consider a smooth exhaustion {Ωk } ↑ M
by open and relatively compact sets of M and the solution of
(
−∆wk + qwk = q on Ωk ,
(12)
wk = 0
on ∂Ωk ,
where q ∈ Cc∞ (M), q ≥ 0 and q 6≡ 0.
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Splitting theorems and geometric aspects of overdetermined problems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒
The maximum principle yields 0 ≤ wk ≤ 1 on ΩK .
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Splitting theorems and geometric aspects of overdetermined problems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒
The maximum principle yields 0 ≤ wk ≤ 1 on ΩK .
The elliptic regularity theory implies that
(
wk on Ωk ,
φk :=
0
on M \ Ωk ,
(13)
belongs to Cc0,1 (M) and satisfies 0 ≤ φk ≤ 1 on M.
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Splitting theorems and geometric aspects of overdetermined problems
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The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒
The sequence {φk } is non-decreasing on M, since
∀ k ≥ 1,
wk ≤ wk+1
on Ωk ,
(14)
again by the maximum principle (applied to wk − wk+1 ).
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Splitting theorems and geometric aspects of overdetermined problems
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Motivations and examples
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒
Therefore, φk ↑ φ ∈ C 2 (M) such that


−∆φ + qφ = q
0≤φ≤1


φ 6≡ 0.
A. FARINA
on M,
on M,
(15)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
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Overdetermined boundary value problems
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒
Therefore, φk ↑ φ ∈ C 2 (M) such that


−∆φ + qφ = q
0≤φ≤1


φ 6≡ 0.
on M,
on M,
(15)
In particular
(
−∆φ = q(1 − φ) ≥ 0
φ ≥ 0, φ 6≡ 0.
A. FARINA
on M,
on M,
(16)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
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Proof of the Splitting theorem
Symmetry results
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒
Therefore, φk ↑ φ ∈ C 2 (M) such that


−∆φ + qφ = q
0≤φ≤1


φ 6≡ 0.
on M,
on M,
(15)
In particular
(
−∆φ = q(1 − φ) ≥ 0
φ ≥ 0, φ 6≡ 0.
on M,
on M,
(16)
the parabolicity of M gives φ ≡ const., so
φ≡1
on M
A. FARINA
and φk ↑ 1.
(17)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒
Z
M
|∇φk |2 =
Z
|∇wk |2 =
Ωk
Z
qwk (1 − wk ) −→ 0
(18)
Ωk
by Lebesgue’s dominated convergence theorem.
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Splitting theorems and geometric aspects of overdetermined problems
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Motivations and examples
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The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of ⇐=
We can suppose w > 0 on M.
Integrate −∆w ≥ 0 against the test function
Z
0≤
∇w ∇
ψ2
w
Z
dx =
A. FARINA
ψ2
w ,
∇w ∇ψ 2
−
w
Z
ψ ∈ Cc0,1 (M), to get
∇w 2
dx
ψ 2 w (19)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of ⇐=
We can suppose w > 0 on M.
Integrate −∆w ≥ 0 against the test function
Z
0≤
ψ2
w ,
ψ ∈ Cc0,1 (M), to get
Z
Z
∇w 2
ψ2
∇w ∇ψ 2
dx
dx =
− ψ 2 w
w
w Z
Z
Z
∇w 2
∇w ∇ψ 2
ψ∇w ∇ψ
dx ≤
ψ 2 =
2
w
w
w
∇w ∇
A. FARINA
(19)
(20)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of ⇐=
We can suppose w > 0 on M.
Integrate −∆w ≥ 0 against the test function
Z
0≤
ψ2
w ,
ψ ∈ Cc0,1 (M), to get
Z
Z
∇w 2
ψ2
∇w ∇ψ 2
dx
dx =
− ψ 2 w
w
w Z
Z
Z
∇w 2
∇w ∇ψ 2
ψ∇w ∇ψ
dx ≤
ψ 2 =
2
w
w
w
Z
Z
12
∇w 2 12 2
dx
≤2
ψ 2 |∇ψ|
dx
w ∇w ∇
A. FARINA
(19)
(20)
(21)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of ⇐=
Z
Z
∇w 2
ψ dx ≤ 4 |∇ψ|2 dx
w 2
A. FARINA
(22)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of ⇐=
Z
Z
∇w 2
ψ dx ≤ 4 |∇ψ|2 dx
w 2
(22)
Insert into (22) ψ = φk , where (φn ) is as in the theorem, to get
Z ∇w 2
w dx = 0
=⇒
w ≡ const.
(23)
M
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of ⇐=
Z
Z
∇w 2
ψ dx ≤ 4 |∇ψ|2 dx
w 2
(22)
Insert into (22) ψ = φk , where (φn ) is as in the theorem, to get
Z ∇w 2
w dx = 0
=⇒
w ≡ const.
(23)
M
Using standard cut-off functions, the same argument gives a simple
proof of a sufficient condition for parabolicity
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
A sufficient condition for parabolicity
Corollary (Cheng, Yau, 1975)
If, for some x0 ∈ M and for a sequence {Rk } ↑ ∞,
Vol(BRk (x0 )) ≤ const.Rk2 ,
(24)
then M is parabolic.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Examples
A compact M is parabolic.
the flat cylinder S1 × R is parabolic.
(R2 , h , ican ) is parabolic.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Examples
A compact M is parabolic.
the flat cylinder S1 × R is parabolic.
(R2 , h , ican ) is parabolic.
for N ≥ 3, (RN , h , ican ) is not parabolic.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Examples
A compact M is parabolic.
the flat cylinder S1 × R is parabolic.
(R2 , h , ican ) is parabolic.
for N ≥ 3, (RN , h , ican ) is not parabolic.
pN(N − 2) N−2
2
u(x) =
1 + |x|2
is a smooth, positive, bounded and non constant solution of
N+2
−∆u = u N−2 ≥ 0
A. FARINA
on RN .
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Sharpness
The function u(x, y ) = y − x 2 satisfies
(
−∆u = 2 on R2 ,
∂u
2
∂y > 0 on R ,
A. FARINA
(25)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Sharpness
The function u(x, y ) = y − x 2 satisfies
(
−∆u = 2 on R2 ,
∂u
2
∂y > 0 on R ,
(25)
(R2 , h , ican ) is parabolic and Ricci-flat,
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Sharpness
The function u(x, y ) = y − x 2 satisfies
(
−∆u = 2 on R2 ,
∂u
2
∂y > 0 on R ,
(25)
(R2 , h , ican ) is parabolic and Ricci-flat,
u is stable, non constant but neither ∇u ∈ L∞ nor
Z
|∇u|2 dx = o(R 2 log R)
as R → +∞.
(26)
BR
are satisfied and u is not 1-D.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
A characterisation of stable solutions
Theorem
Let M be a complete, Riemannian manifold without boundary. Then
u stable ⇐⇒ there is a smooth function w such that :
(
−∆w − f 0 (u)w ≥ 0 on M
w >0
on M
(27)
[Moss, Piepenbrink (1978), Fischer-Colbrie, Schoen (1980)]
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒ : the compact case
Let (ϕ1 , λ1 ) the first eigenpair for L := −∆ − f 0 (u)Id, then
(
ϕ1 > 0 on M
λ1 ≥ 0 on M
(28)
by the stability assumption (7) and the variational characterisation
of the first eigenvalue. Thus
− ∆ϕ1 − f 0 (u)ϕ1 = λ1 ϕ1 ≥ 0
A. FARINA
on M.
(29)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒ : the non-compact case
Let (ϕR , λR1 ) the first eigenpair for
(
−∆ϕR − f 0 (u)ϕR = λR1 ϕR
ϕR = 0
on BR (x0 )
on ∂BR (x0 )
(30)
where x0 ∈ M and BR (x0 ) := {x ∈ M : dist(x, x0 ) < R}.
The eigenfunctions can be chosen in such a way that ϕR > 0 on
BR (x0 ) and normalized so that ϕR (x0 ) = 1.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒ : the non-compact case
The stability assumption (7) and the variational characterisation of
the first eigenvalue imply
∀R>0
A. FARINA
λR1 ≥ 0.
(31)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒ : the non-compact case
The stability assumption (7) and the variational characterisation of
the first eigenvalue imply
∀R>0
λR1 ≥ 0.
(31)
Also (again by the variational characterisation of the first eigenvalue)
λR1 ↓ λ,
(32)
for some λ ≥ 0.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒ : the non-compact case
By applying Harnack inequality to
(
−∆ϕR − (f 0 (u) + λR1 )ϕR = 0 on BR (x0 )
ϕR = 0
on ∂BR (x0 ).
(33)
we have : ∀ Bσ (x0 ), ∃ C = C (σ, M) > 0, but not depending on R :
∀ R > 2σ
0 < ϕR ≤ C
on Bσ (x0 )
(34)
Here we used that kf 0 (u) + λR1 kL∞ (Bσ (x0 )) is bounded independently
on R.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒ : the non-compact case
By applying Harnack inequality to
(
−∆ϕR − (f 0 (u) + λR1 )ϕR = 0 on BR (x0 )
ϕR = 0
on ∂BR (x0 ).
(33)
we have : ∀ Bσ (x0 ), ∃ C = C (σ, M) > 0, but not depending on R :
∀ R > 2σ
0 < ϕR ≤ C
on Bσ (x0 )
(34)
Here we used that kf 0 (u) + λR1 kL∞ (Bσ (x0 )) is bounded independently
on R.
Standard elliptic estimates and a classical diagonal argument yield
{ϕR }R>0
A. FARINA
2
is compact in Cloc
(M)
(35)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒ : the non-compact case
Therefore, up to a subsequence, ϕR → w ∈ C 2 (M) such that

0

−∆w − f (u)w = λw on M,
w ≥0
on M,


w (x0 ) = 1.
A. FARINA
(36)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of =⇒ : the non-compact case
Therefore, up to a subsequence, ϕR → w ∈ C 2 (M) such that

0

−∆w − f (u)w = λw on M,
w ≥0
on M,


w (x0 ) = 1.
(36)
The strong maximum principle gives
w >0
A. FARINA
on M.
(37)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of ⇐=
2
Integrate −∆w − f 0 (u)w ≥ 0 against the test function ψw , ψ ∈ Cc0,1 (M),
to get
2
Z
Z
ψ
dx − f 0 (u)ψ 2 dx ≥ 0.
(38)
∇w ∇
w
On the other hand
2
2
ψ
ψ 2
2
∇w ∇
= |∇ψ| − w ∇
,
(39)
w
w A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The general Splitting Theorem
The compact case
Parabolicity
Examples
Sharpness
A characterisation of stable solutions
Proof of ⇐=
2
Integrate −∆w − f 0 (u)w ≥ 0 against the test function ψw , ψ ∈ Cc0,1 (M),
to get
2
Z
Z
ψ
dx − f 0 (u)ψ 2 dx ≥ 0.
(38)
∇w ∇
w
On the other hand
2
2
ψ
ψ 2
2
∇w ∇
= |∇ψ| − w ∇
,
(39)
w
w and thus
Z
0≤
2
ψ dx
w 2 ∇
w Z
≤
A. FARINA
|∇ψ|2 dx −
Z
f 0 (u)ψ 2 dx
(40)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Proof of the Splitting theorem
u stable =⇒ there is a smooth function w such that :
(
−∆w − f 0 (u)w ≥ 0 on M
w >0
on M
A. FARINA
(41)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Proof of the Splitting theorem
u stable =⇒ there is a smooth function w such that :
(
−∆w − f 0 (u)w ≥ 0 on M
w >0
on M
Integrate ∆w + f 0 (u)w ≤ 0 against the test function
ψ ∈ Cc0,1 (M), to deduce :
A. FARINA
(41)
ψ2
w ,
Splitting theorems and geometric aspects of overdetermined problems
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Proof of the Splitting theorem
u stable =⇒ there is a smooth function w such that :
(
−∆w − f 0 (u)w ≥ 0 on M
w >0
on M
Integrate ∆w + f 0 (u)w ≤ 0 against the test function
ψ ∈ Cc0,1 (M), to deduce :
Z
2
ψ w ∇
dx
w 2
Z
≤
A. FARINA
2
|∇ψ| dx −
Z
(41)
ψ2
w ,
f 0 (u)ψ 2 dx
(42)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Next we use the Böchner formula :
1
∆|∇u|2 = h∇∆u, ∇ui + Ric(∇u, ∇u) + |∇du|2 .
2
A. FARINA
(43)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Next we use the Böchner formula :
1
∆|∇u|2 = h∇∆u, ∇ui + Ric(∇u, ∇u) + |∇du|2 .
2
(43)
Since u solves −∆u = f (u) on M, the Böchner formula gives :
1
∆|∇u|2 = −f 0 (u)|∇u|2 + Ric(∇u, ∇u) + |∇du|2 .
2
A. FARINA
(44)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Next we use the Böchner formula :
1
∆|∇u|2 = h∇∆u, ∇ui + Ric(∇u, ∇u) + |∇du|2 .
2
(43)
Since u solves −∆u = f (u) on M, the Böchner formula gives :
1
∆|∇u|2 = −f 0 (u)|∇u|2 + Ric(∇u, ∇u) + |∇du|2 .
(44)
2
Integrating the latter against the test function φ2 , φ ∈ Cc0,1 (M), we
get :
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Next we use the Böchner formula :
1
∆|∇u|2 = h∇∆u, ∇ui + Ric(∇u, ∇u) + |∇du|2 .
2
(43)
Since u solves −∆u = f (u) on M, the Böchner formula gives :
1
∆|∇u|2 = −f 0 (u)|∇u|2 + Ric(∇u, ∇u) + |∇du|2 .
(44)
2
Integrating the latter against the test function φ2 , φ ∈ Cc0,1 (M), we
get :
Z
|∇du|2 + Ric(∇u, ∇u) φ2 dx
Z
=
f 0 (u)|∇u|2 φ2 dx −
A. FARINA
Z
φh∇φ, ∇|∇u|2 idx.
(45)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Use the spectral inequality (42) with test function ψ = φ|∇u| ∈ Cc0,1 (M),
to get
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Use the spectral inequality (42) with test function ψ = φ|∇u| ∈ Cc0,1 (M),
to get
Z h
i
2
|∇du|2 − ∇|∇u| + Ric(∇u, ∇u) φ2 dx
Z
≤
2
2
|∇φ| |∇u| dx −
A. FARINA
Z
2
φ|∇u| w ∇
dx.
w
(46)
2
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Z h
Z
+ (1 − δ)
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
i
2
|∇du|2 − ∇|∇u| + Ric(∇u, ∇u) φ2 dx
2
Z
|∇u| 1
φ w ∇
dx
≤
|∇φ|2 |∇u|2 dx
w
δ
2
2
(47)
for some 0 < δ < 1.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
The parabolic case
In case (i), i.e. when M is parabolic, we plug into (47) the sequence of
test functions given by (11) to obtain, when n → +∞
Z h
i
2
|∇du|2 − ∇|∇u| + Ric(∇u, ∇u) dx
Z
+ (1 − δ)
2
|∇u| w ∇
dx ≤ 0
w
2
(48)
for some 0 < δ < 1.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
The parabolic case
In case (i), i.e. when M is parabolic, we plug into (47) the sequence of
test functions given by (11) to obtain, when n → +∞
Z h
i
2
|∇du|2 − ∇|∇u| + Ric(∇u, ∇u) dx
Z
+ (1 − δ)
2
|∇u| w ∇
dx ≤ 0
w
2
(48)
for some 0 < δ < 1.
Which leads to
|∇u| = cw , for some c ≥ 0,
2
|∇du|2 = ∇|∇u| , Ric(∇u, ∇u) = 0.
A. FARINA
(49)
(50)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
The non-parabolic case
In case (ii), we apply a logarithmic cutoff argument.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
The non-parabolic case
In case (ii), we apply a logarithmic cutoff argument.
For fixed R > 0, choose the following radial φ(x) = φR (r (x)):
√

1
if r ≤ R,



√
log r
φR (r ) =
2
−
2
if
r
∈
[
R, R],

log R


0
if r ≥ R.
A. FARINA
(51)
Splitting theorems and geometric aspects of overdetermined problems
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
The non-parabolic case
Note that
|∇φ(x)|2 =
4
χBR \B√R (x),
r (x)2 log2 R
Choose R in such a way that log2 R is an integer. Then,
Z
Z
|∇φ|2 |∇u|2 dx =
|∇φ|2 |∇u|2 dx
BR \B√R
M
=
≤
4
log2 R
4
log2 R
logX
R−1
k=log R/2
log
XR
k=log R/2
Z
Be k+1 \Be k
1
e 2k
A. FARINA
Z
|∇u|2
dx
r (x)2
(52)
|∇u|2 dx.
Be k+1
Splitting theorems and geometric aspects of overdetermined problems
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
The non-parabolic case
By assumption,
Z
|∇u|2 dx ≤ (k + 1)e 2(k+1) δ(k)
Be k+1
for some δ(k) satisfying δ(k) → 0 as k → +∞.
W.l.o.g., we can assume δ(k) to be decreasing as a function of k. So
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Splitting theorems and geometric aspects of overdetermined problems
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
The non-parabolic case
By assumption,
Z
|∇u|2 dx ≤ (k + 1)e 2(k+1) δ(k)
Be k+1
for some δ(k) satisfying δ(k) → 0 as k → +∞.
W.l.o.g., we can assume δ(k) to be decreasing as a function of k. So
4
log2 R
log
XR
k=log R/2
2
1
e 2k
Z
Be k+1
8
|∇u| dx ≤
log2 R
2
log
XR
k=log R/2
e 2(k+1)
(k + 1)δ(k)
e 2k
log
XR
8e
C
(k + 1) ≤
δ(log R/2)
δ(log R/2) log2 R = C δ(log R/2),
2
2
log R
log
R
k=0
(53)
for some constant C > 0, independent of R.
≤
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
The non-parabolic case
By assumption,
Z
|∇u|2 dx ≤ (k + 1)e 2(k+1) δ(k)
Be k+1
for some δ(k) satisfying δ(k) → 0 as k → +∞.
W.l.o.g., we can assume δ(k) to be decreasing as a function of k. So
4
log2 R
log
XR
k=log R/2
2
1
e 2k
Z
Be k+1
8
|∇u| dx ≤
log2 R
2
log
XR
k=log R/2
e 2(k+1)
(k + 1)δ(k)
e 2k
log
XR
8e
C
(k + 1) ≤
δ(log R/2)
δ(log R/2) log2 R = C δ(log R/2),
2
2
log R
log
R
k=0
(53)
for some constant C > 0, independent of R.
≤
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
To conclude we plug the cut-off functions φR into (47) to get
|∇u| = cw , for some c ≥ 0,
2
|∇du|2 = ∇|∇u| , Ric(∇u, ∇u) = 0.
A. FARINA
(54)
(55)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
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References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
To conclude we plug the cut-off functions φR into (47) to get
|∇u| = cw , for some c ≥ 0,
2
|∇du|2 = ∇|∇u| , Ric(∇u, ∇u) = 0.
(54)
(55)
Since u is non-constant by assumption, we must have c > 0 and thus
|∇u| > 0
A. FARINA
on M.
(56)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
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References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Lemma
Let u ∈ C 2 be a function on M, and let p ∈ M be a point such that
∇u(p) 6= 0. Then, denoting with II the second fundamental form of the
level set Σ = {u = u(p)} in a neighbourhood of p, it holds
2
2
|∇du|2 − ∇|∇u| = |∇u|2 |II |2 + ∇T |∇u| ,
where ∇T is the tangential gradient on the level set Σ.
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Proof of SZ
Fix a local orthonormal frame {ei } on Σ, and let ν = ∇u/|∇u| be the
normal vector. For every vector field X ∈ Γ(TM),
∇du(ν, X ) =
1
1
∇du(∇u, X ) =
h∇|∇u|2 , X i = h∇|∇u|, X i.
|∇u|
2|∇u|
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Proof of SZ
Fix a local orthonormal frame {ei } on Σ, and let ν = ∇u/|∇u| be the
normal vector. For every vector field X ∈ Γ(TM),
∇du(ν, X ) =
1
1
∇du(∇u, X ) =
h∇|∇u|2 , X i = h∇|∇u|, X i.
|∇u|
2|∇u|
Moreover, for a level set
II = −
∇du|T Σ×T Σ
.
|∇u|
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Proof of SZ
Fix a local orthonormal frame {ei } on Σ, and let ν = ∇u/|∇u| be the
normal vector. For every vector field X ∈ Γ(TM),
∇du(ν, X ) =
1
1
∇du(∇u, X ) =
h∇|∇u|2 , X i = h∇|∇u|, X i.
|∇u|
2|∇u|
Moreover, for a level set
II = −
|∇du|2
=
∇du|T Σ×T Σ
.
|∇u|
X
X
2 2
2
∇du(ei , ej ) + 2
∇du(ν, ej ) + ∇du(ν, ν)
j
X
2 2
= |∇u| |II | + 2
h∇|∇u|, ej i + h∇|∇u|, νi
i,j
2
2
j
2 2
= |∇u|2 |II |2 + ∇T |∇u| + ∇|∇u| .
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
By summarizing, we have proven
|∇u| > 0,
2
|∇du|2 = ∇|∇u| , Ric(∇u, ∇u) = 0,
2
|II |2 = 0, ∇T |∇u| = 0 on Σ.
A. FARINA
(57)
(58)
(59)
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
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References
Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
The differentiable splitting is obtained since u is without critical
points and |∇u| is constant on the level sets of u.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
The differentiable splitting is obtained since u is without critical
points and |∇u| is constant on the level sets of u.
The flow Φ of the unit vector field ν = ∇u/|∇u| is well defined on
M × R and, since |∇u| is constant on the level sets of u, Φt moves
level sets of u onto level sets of u. Therefore, having chosen a level
set N, the map Φ : N × R → M is a diffeomorphism.
Also N is totally geodesic.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Since |∇u| is constant on level sets of u, |∇u| = α(u) for some function
α. Evaluating along curves Φt (x), since u ◦ Φt is a local bijection we
deduce that α is continuous.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Since |∇u| is constant on level sets of u, |∇u| = α(u) for some function
α. Evaluating along curves Φt (x), since u ◦ Φt is a local bijection we
deduce that α is continuous.
Claim : Φt moves level sets of u to level sets of u.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Since |∇u| is constant on level sets of u, |∇u| = α(u) for some function
α. Evaluating along curves Φt (x), since u ◦ Φt is a local bijection we
deduce that α is continuous.
Claim : Φt moves level sets of u to level sets of u.
Observe that
d
(u ◦ γ) = h∇u, νi = |∇u| ◦ γ > 0,
dt
where γ is any integral curve of ν.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Since |∇u| is constant on level sets of u, |∇u| = α(u) for some function
α. Evaluating along curves Φt (x), since u ◦ Φt is a local bijection we
deduce that α is continuous.
Claim : Φt moves level sets of u to level sets of u.
Observe that
d
(u ◦ γ) = h∇u, νi = |∇u| ◦ γ > 0,
dt
where γ is any integral curve of ν.
Therefore, integrating d/ds(u ◦ Φs ) = |∇u| ◦ Φs = α(u ◦ Φs ) we get
Z
u(Φt (x))
t=
u(x)
dξ
,
α(ξ)
thus u(Φt (x)) is independent of x varying in a level set. As α(ξ) > 0,
this also shows that flow lines starting from a level set of u do not touch
the same level set.
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
Injectivity of Φ|N×R .
Suppose that Φ(x1 , t1 ) = Φ(x2 , t2 ). Then, since Φ moves level sets to
level sets, necessarily t1 = t2 = t. If by contradiction x1 6= x2 , two
distinct flow lines of Φt would intersect at the point Φt (x1 ) = Φt (x2 ),
contradicting the fact that Φt is a diffeomorphism on M for every t.
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
• For the Riemannian part of the splitting we consider the Lie derivative
of the metric in the direction of Φt and prove that Lν h , i = 0, thus Φt is
a flow of isometries.
Lν h , i (X , Y ) = h∇X ν, Y i + hX , ∇Y νi
=
+
2
∇du(X , Y ) + X
|∇u|
1
h∇u, X i.
Y |∇u|
1
|∇u|
h∇u, Y i
From the expression, using that |∇u| is constant on N and the properties
of ∇du we deduce that
Lν h , i (X , Y ) =
2
∇du(X , Y ) = 0.
|∇u|
If at least one between X and Y is in the tangent space of N.
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
If, however, X and Y are normal, (w.l.o.g. X = Y = ∇u), we have
2
1
Lν h , i (∇u, ∇u) =
∇du(∇u, ∇u) + 2∇u
|∇u|2
|∇u|
|∇u|
2
=
∇du(∇u, ∇u) − 2∇u(|∇u|)
|∇u|
=
2∇du(ν, ∇u) − 2h∇|∇u|, ∇ui = 0.
Concluding, Lν h , i = 0, thus Φt is a flow of isometries. Since ∇u ⊥ TN,
M splits as a Riemannian product, as desired.
In particular, RicN ≥ 0 if m ≥ 3, while, if m = 2, M = R2 or S1 × R with
the flat metric.
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Splitting theorems and geometric aspects of overdetermined problems
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
• In a local Darboux frame {ej , ν} for the level surface N,
0 = |II |2 =⇒ ∇du(ei , ej ) = 0
0 = h∇|∇u|, ej i = ∇du(ν, ej ),
(60)
so the unique nonzero component of ∇du is that corresponding to the
pair (ν, ν). Also recall that
d
(u ◦ γ) = h∇u, νi = |∇u| ◦ γ > 0,
dt
where γ is any integral curve of ν. Hence
A. FARINA
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Proof
The parabolic case
The non-parabolic case
SZ
Differentiable splitting
Level sets
Injectivity
Riemannian part of the splitting
ODE
• In a local Darboux frame {ej , ν} for the level surface N,
0 = |II |2 =⇒ ∇du(ei , ej ) = 0
0 = h∇|∇u|, ej i = ∇du(ν, ej ),
(60)
so the unique nonzero component of ∇du is that corresponding to the
pair (ν, ν). Also recall that
d
(u ◦ γ) = h∇u, νi = |∇u| ◦ γ > 0,
dt
where γ is any integral curve of ν. Hence
−f (u ◦ γ)
=
∆u(γ) = ∇du(ν, ν)(γ) = h∇|∇u|, νi(γ)
d
d2
(|∇u| ◦ γ) = 2 (u ◦ γ),
dt
dt
thus y = u ◦ γ solves the ODE −y 00 = f (y ).
=
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
De Giorgi’s conjecture
In 1978 E. De Giorgi formulated the following question :
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
De Giorgi’s conjecture
In 1978 E. De Giorgi formulated the following question :
Let u ∈ C 2 (Rm , [−1, 1]) satisfy
− ∆u = u − u 3
and
A. FARINA
∂u
>0
∂xm
on Rm .
(61)
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
De Giorgi’s conjecture
In 1978 E. De Giorgi formulated the following question :
Let u ∈ C 2 (Rm , [−1, 1]) satisfy
− ∆u = u − u 3
and
∂u
>0
∂xm
on Rm .
(61)
Is it true that all the level sets of u are hyperplanes, at least if m ≤ 8?
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
De Giorgi’s conjecture
In 1978 E. De Giorgi formulated the following question :
Let u ∈ C 2 (Rm , [−1, 1]) satisfy
− ∆u = u − u 3
and
∂u
>0
∂xm
on Rm .
(61)
Is it true that all the level sets of u are hyperplanes, at least if m ≤ 8?
The original conjecture has been proven in dimensions m = 2, 3 and it is
still open, in its full generality, for 4 ≤ m ≤ 8.
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
• m = 2 (Ghoussoub-Gui (1998)),
• m = 3 (Ambrosio-Cabré (2000)).
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
• m = 2 (Ghoussoub-Gui (1998)),
• m = 3 (Ambrosio-Cabré (2000)).
When 4 ≤ m ≤ 8, the conjecture is true under additional assumptions :
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
• m = 2 (Ghoussoub-Gui (1998)),
• m = 3 (Ambrosio-Cabré (2000)).
When 4 ≤ m ≤ 8, the conjecture is true under additional assumptions :
• (Savin (2009)),
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Splitting theorems and geometric aspects of overdetermined problems
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
• m = 2 (Ghoussoub-Gui (1998)),
• m = 3 (Ambrosio-Cabré (2000)).
When 4 ≤ m ≤ 8, the conjecture is true under additional assumptions :
• (Savin (2009)),
• (A.F. - Valdinoci (2010)).
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Splitting theorems and geometric aspects of overdetermined problems
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
• m = 2 (Ghoussoub-Gui (1998)),
• m = 3 (Ambrosio-Cabré (2000)).
When 4 ≤ m ≤ 8, the conjecture is true under additional assumptions :
• (Savin (2009)),
• (A.F. - Valdinoci (2010)).
• m ≥ 9, (Del Pino, Kowalczyk, Wei (2011)) have constructed a solution
satisfying
∂u
|u| ≤ 1,
>0
∂xN
which is not a one-dimensional function.
This implies that the assumption m ≤ 8 in De Giorgi’s conjecture cannot
be removed.
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
X = ∂/∂xm is a Killing field on (Rm , h , ican )
and
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
X = ∂/∂xm is a Killing field on (Rm , h , ican )
and
Let X be a Killing vector field on (M m , h , i) and let u ∈ C 3 (M) be a
solution of −∆u = f (u), for some f ∈ C 1 (R). Then, the function
w = h∇u, X i solves
∆w + f 0 (u)w = 0
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
Theorem
Let (M, h , i) be a complete non-compact Riemannian manifold without
boundary with Ric ≥ 0 and let X be a Killing field on M. Suppose
f ∈ C 1 (R) and u ∈ C 3 (M) such that
(
−∆u = f (u)
on M,
h∇u, X i > 0
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
Theorem
Let (M, h , i) be a complete non-compact Riemannian manifold without
boundary with Ric ≥ 0 and let X be a Killing field on M. Suppose
f ∈ C 1 (R) and u ∈ C 3 (M) such that
(
−∆u = f (u)
on M,
h∇u, X i > 0
on M,
If either
(i) M is parabolic and ∇u ∈ L∞ (M) or
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
Theorem
(ii) the function |∇u| satisfies
Z
|∇u|2 dx = o(R 2 log R)
as R → +∞,
BR
then,
M = N × R with the product metric h , i = h , iN + dt 2 , for some
complete, totally geodesic, parabolic submanifold N with RicN ≥ 0.
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Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
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De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
Theorem
(ii) the function |∇u| satisfies
Z
|∇u|2 dx = o(R 2 log R)
as R → +∞,
BR
then,
M = N × R with the product metric h , i = h , iN + dt 2 , for some
complete, totally geodesic, parabolic submanifold N with RicN ≥ 0.
In particular,
M = R2 or S1 × R, with their flat metric, if m = 2;
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Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
Theorem
(ii) the function |∇u| satisfies
Z
|∇u|2 dx = o(R 2 log R)
as R → +∞,
BR
then,
M = N × R with the product metric h , i = h , iN + dt 2 , for some
complete, totally geodesic, parabolic submanifold N with RicN ≥ 0.
In particular,
M = R2 or S1 × R, with their flat metric, if m = 2;
u depends only on t and writing u = y (t) it holds that
−y 00 = f (y ),
A. FARINA
y 0 > 0.
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
De Giorgi’s conjecture
An extended version of De Giorgi’s conjecture
Corollary
Let (M, h , i) be a complete non-compact surface without boundary, with
Gaussian curvature K ≥ 0 and let X be a Killing field on M. Suppose
that u ∈ C 3 (M) is a solution of

on M

 −∆u = f (u)
h∇u, X i > 0
on M


∞
∇u ∈ L (M)
with f ∈ C 1 (R).
Then, M is the Riemannian product R2 or S1 × R, with flat metric, u
depends only on t and, writing u = y (t), it holds
−y 00 = f (y ),
A. FARINA
y 0 > 0.
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Serrin’s problem in unbounded domains
In 1997, H. Berestycki, L. Caffarelli and L. Nirenberg, motivated by
questions on the regularity of some one-phase free boundary problems,
are led to the study of semilinear problems of bistable type in globally
Lipschitz unbounded domains.
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Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Serrin’s problem in unbounded domains
In 1997, H. Berestycki, L. Caffarelli and L. Nirenberg, motivated by
questions on the regularity of some one-phase free boundary problems,
are led to the study of semilinear problems of bistable type in globally
Lipschitz unbounded domains.
They considered the case of a smooth, globally Lipschitz epigraph,
i.e. a domain Ω of the form :
0
0
Ω := { (x , xN ) ∈ RN : ϕ(x ) < xN },
where ϕ : RN−1 → R is a globally Lipschitz smooth function.
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Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
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Overdetermined boundary value problems
References
Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Serrin’s problem in unbounded domains
In particular, they consider the following overdetermined b.v.p.


in Ω
−∆u = f (u)
u>0
in Ω


∂u
u = 0, ∂ν = const. on ∂Ω
(63)
where Ω is a globally lipschitz smooth epigraph of RN and f is of bistable
type.
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Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
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Overdetermined boundary value problems
References
Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Serrin’s problem in unbounded domains
In particular, they consider the following overdetermined b.v.p.


in Ω
−∆u = f (u)
u>0
in Ω


∂u
u = 0, ∂ν = const. on ∂Ω
(63)
where Ω is a globally lipschitz smooth epigraph of RN and f is of bistable
type.
They ask, in analogy with Serrin’s theorem,
Question
If the overdetermined b.v.p. (63) admits a bounded C 2 (Ω)-solution, is it
true that Ω is a half-space ?
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Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
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The framework
Splitting theorems
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Overdetermined boundary value problems
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Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Theorem (F. - Valdinoci, 2010)
Let f ∈ C 1 be of bistable type and Ω a globally lipschitz smooth epigraph
of RN , with N = 2, 3. If the overdetermined boundary value problem


in Ω
−∆u = f (u)
(64)
u>0
in Ω


∂u
u = 0, ∂ν = const. on ∂Ω
admits a bounded C 2 (Ω)-solution,
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Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
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Overdetermined boundary value problems
References
Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Theorem (F. - Valdinoci, 2010)
Let f ∈ C 1 be of bistable type and Ω a globally lipschitz smooth epigraph
of RN , with N = 2, 3. If the overdetermined boundary value problem


in Ω
−∆u = f (u)
(64)
u>0
in Ω


∂u
u = 0, ∂ν = const. on ∂Ω
admits a bounded C 2 (Ω)-solution, then up to isometry Ω is the
N−1
half-space RN
× (0, +∞) and u is one-dimensional and
+ := R
∂u
monotone (that is u = u(xN ) and ∂x
> 0).
N
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Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
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Overdetermined boundary value problems
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Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Theorem (F. - Valdinoci, 2010)
Let f ∈ C 1 be of bistable type and Ω a globally lipschitz smooth epigraph
of RN , with N = 2, 3. If the overdetermined boundary value problem


in Ω
−∆u = f (u)
(64)
u>0
in Ω


∂u
u = 0, ∂ν = const. on ∂Ω
admits a bounded C 2 (Ω)-solution, then up to isometry Ω is the
N−1
half-space RN
× (0, +∞) and u is one-dimensional and
+ := R
∂u
monotone (that is u = u(xN ) and ∂x
> 0).
N
A crucial step is to prove that
A. FARINA
∂u
∂xN
> 0 in Ω.
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Overdetermined bvp in a Riemannian setting
Theorem
Let (M, h , i) be a complete, non-compact Riemannian manifold without
boundary, satisfying Ric ≥ 0 and let X be a Killing field on M. Let
Ω ⊆ M be an open and connected set with C 3 boundary. Suppose that
u ∈ C 3 (Ω) is a non-constant solution of the overdetermined problem

on Ω

 −∆u = f (u)
u = constant
on ∂Ω
(65)


∂ν u = constant 6= 0
on ∂Ω
such that h∇u, X i is either positive or negative on Ω.
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Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
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Overdetermined boundary value problems
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Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Theorem
Then, if either
(i) M is parabolic and ∇u ∈ L∞ (Ω), or
(ii) the function |∇u| satisfies
Z
|∇u|2 dx = o(R 2 log R)
as R → +∞,
Ω∩BR
the following properties hold true:
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Splitting theorems and geometric aspects of overdetermined problems
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Splitting theorems
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Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Theorem
Then, if either
(i) M is parabolic and ∇u ∈ L∞ (Ω), or
(ii) the function |∇u| satisfies
Z
|∇u|2 dx = o(R 2 log R)
as R → +∞,
Ω∩BR
the following properties hold true:
Ω = ∂Ω × R+ with the product metric h , i = h , i∂Ω + dt 2 , ∂Ω is
totally geodesic in M and satisfies Ric∂Ω ≥ 0,
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Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
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Overdetermined boundary value problems
References
Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Theorem
Then, if either
(i) M is parabolic and ∇u ∈ L∞ (Ω), or
(ii) the function |∇u| satisfies
Z
|∇u|2 dx = o(R 2 log R)
as R → +∞,
Ω∩BR
the following properties hold true:
Ω = ∂Ω × R+ with the product metric h , i = h , i∂Ω + dt 2 , ∂Ω is
totally geodesic in M and satisfies Ric∂Ω ≥ 0,
the function u depends only on t, it has no critical points, and
writing u = y (t) it holds −y 00 = f (y ).
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Idea of the proof
Z h
i
2
|∇du|2 − ∇|∇u| + Ric(∇u, ∇u) φ2 dx
Ω
2
φ|∇u| ≤
|∇φ| |∇u| dx −
w ∇
dx.
w
Ω
Ω
Z
2
2
Z
(66)
2
For every φ ∈ Cc0,1 (M) and not only in Cc0,1 (Ω).
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Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Idea of the proof
Z h
i
2
|∇du|2 − ∇|∇u| + Ric(∇u, ∇u) φ2 dx
Ω
2
φ|∇u| ≤
|∇φ| |∇u| dx −
w ∇
dx.
w
Ω
Ω
Z
2
Z
2
(66)
2
For every φ ∈ Cc0,1 (M) and not only in Cc0,1 (Ω).
Here we crucially use the overdetermined conditions
u = 0,
∂u
= const.
∂ν
A. FARINA
on
∂Ω
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
Serrin’s problem in unbounded domains
Overdetermined bvp in a Riemannian setting
Thank you for your
attention
A. FARINA
Splitting theorems and geometric aspects of overdetermined problems
Introduction and aims
Motivations and examples
The framework
Splitting theorems
Proof of the Splitting theorem
Symmetry results
Overdetermined boundary value problems
References
References
The lectures are based on
Splitting theorems, symmetry results and overdetermined problems
for Riemannian manifolds, Comm. in P.D.E., vol. 38, Issue 10, 2013
(with L. Mari and E. Valdinoci).
Flattening results for elliptic PDEs in unbounded domains with
applications to overdetermined problems, Archive for Rational
Mechanics and Analysis, 195 (2010), 1025 - 1058. (with E.
Valdinoci).
1D symmetry for solutions of semilinear and quasilinear elliptic
equations, Trans. of the AMS, Volume 363, Number 2, February
2011, Pages 579 - 609 (with E. Valdinoci).
The state of the art for a conjecture of De Giorgi and related
problems, in Recent Progress on Reaction-Diffusion Systems and
Viscosity Solutions, March 2009. Edited by H.Ishii, W.-Y.Lin and
Y.Du, World Scientific (with E. Valdinoci).
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Splitting theorems and geometric aspects of overdetermined problems