A min-max formula for operators that have the global
comparison principle (i.e. elliptic operators)
Russell Schwab (Michigan State University)
Bedlewo Conference on Nonlocal Operators and PDE
29 June 2016
Collaboration
This is joint work with Nestor Guillen
Warm-up Discussion
What are your top 5 favorite ways to define the (1/2)-Laplacian (or
s-Laplacian)?
Warm-up Discussion
Assume φ is C 1,γ (Rd ) and Uφ is the unique bounded solution of
(
∆Uφ = 0
Uφ = φ
in Rd+1
+
on Rd .
It is well known that
φ 7→ ∂n Uφ = −(−∆)1/2 φ on Rd
−(−∆)1/2 φ(x) = p.v .
Z
Rd
(u(x + h) − u(x)) |h|−d−1 dh
Warm-up Discussion
Let Ω ⊂ Rd . Given φ ∈ C 1,α (∂Ω) (α > 0), let Uφ solve
(
F (D 2 Uφ , x) = 0 in Ω
Uφ = φ
on ∂Ω
where F is a uniformly elliptic operator and Uφ is the unique viscosity
solution. Set
I (φ, x) := ∂n Uφ (x)
( n = inner normal to ∂Ω)
Z
I (φ, x)“ = ”nonlinear version
∂Ω
(u(y ) − u(x)) |y − x|−d−1 dσ(h)??????
Warm-up Discussion
How many papers can you list that start with a fractional problem e.g.
involving −(−∆)s and resolve it by extending to an extra dimension?
How many papers can you list that start with a second order problem
and resolve it by fractional (or integro-differential) techniques in one
LESS dimension?
Linear Lévy Operator
L : C 2 → L∞ , with u ∈ C 2
Lloc (u,x)
}|
{
z
L(u, x) = Tr(A(x)D 2 u) + B(x) · ∇u + C (x)u +
Z
(u(x + y ) − u(x) − y · ∇u(x)1B1 (y ))dµx (y )
d
|R
{z
}
LID (u,x)
where A(x) ≥ 0, C (x) ≤ 0, all of A, B, C are bounded, and µx satisfies
Z
min(|y |2 , 1)dµx (y ) < +∞.
Rd
Linear Lévy Operator
Generators of Markov processes on Rd
Ex (u(Xtx (·))) − u(x)
t→0
t
L(u, x) := lim
The Global Comparison Principle
Definition
I : D ⊂ RX → RX
is said to have the global comparison property (GCP) if
u, v ∈ D and v touches u from above at x0 ⇒ I (u, x0 ) ≤ I (v , x0 )
u(x) ≤ v (x) ∀x ∈ X
u(x0 ) = v (x0 )
Examples of the GCP
Consider X = Rd , D = C 2 (Rd )
• I (f , x) = ∆f (x)
• I (f , x) = Tr(A(x)D 2 f (x))) + B(x) · ∇f , A(x) ≥ 0
• I (f , x) = F (D 2 f (x)), F : Sym(Rd ) → R monotone nondec.
• I (f , x) = H(∇f (x)), H : Rd → Rd , D = C 1 (Rd ))
• I (f , x) = f (x + y ) − f (x), y ∈ Rd fixed, (D = C 0 (Rd ))
• I (f , x) =
R
f (x + y ) − f (x)dµ(y ), µ Borel measure (D = C 0 (Rd ))
• I (f , x) = −(−∆)s f (x), s ∈ [0, 1], (D = S(Rd )), dµ(y ) = |y |−d−2s
Examples of the GCP
Dirichlet to Neumann map (D-to-N) for nonlinear elliptic equations
Let X = ∂Ω, Ω ⊂ Rd . Given φ ∈ C 1,α (∂Ω) (α > 0), let Uφ solve
(
F (D 2 Uφ , x) = 0 in Ω
Uφ = φ
on ∂Ω
where F is a uniformly elliptic operator and Uφ is the unique viscosity
solution. Set
I (φ, x) := ∂n Uφ (x)
( n = inner normal to ∂Ω)
The map I : C 1,α (∂Ω) → C (∂Ω) is Lipschitz and has the GCP.
Linear GCP Characterization
Theorem (Philippe Courrège 1965)
Suppose L is an operator for which
1. L : C02 (Rd ) → C (Rd )
2. L is linear
3. L satisfies the GCP.
Then L is necessarily a linear Lévy operator,
L(u, x) = Tr(A(x)D 2 u)+B(x) · ∇u + C (x)u+
Z
(u(x + y ) − u(x) − y · ∇u(x)1B1 (y ))dµx (y )
Rd
where A(x) ≥ 0, C (x) ≤ 0, all of A, B, C are bounded, and µx satisfies
Z
min(y 2 , 1)dµx (y ) < +∞.
Rd
Lipschitz maps with the GCP
Many of the interesting examples, even in the preceding discussion were
not linear– especially the D-to-N. Can you still prove a similar
characterization as what Courrège proved for linear operators?
Yes, if you assume I is Lipschitz
GCP is a lot of structure!
For example...
|I (u, x) − I (v , x)| ≤
C (R)kI kLip(C β ,C ) (ku − v kC β (BR (x)) + ku − v kL∞ (M) ).
b
b
A new result
Theorem (Guillen-Schwab, arXiv two days ago!)
Let M be a complete, d-dimensional manifold, and that
I : C 2 (M) → C 0 (M) is Lipschitz, with the GCP. Then
I (u, x) = min max{fab (x) + Lab (u, x)} ∀ u, x.
a
b
where, for each pair of indices ab, we have
• fab (x) ∈ C 0 (Rd ) (uniformly)
• Lab : C 2 (Rd ) → C 0 (R) is a (uniformly) bounded linear Lévy
operator
A new result
Theorem (Guillen-Schwab, arXiv two days ago!)
Furthermore if I : C 1,γ (Rd ) → C (Rd ) is Lipschitz and satisfies the GCP,
then
L(u, x) = C (x)u(x) + B(x) · ∇u
Z
+
u(x + y ) − u(x) − ∇u(x) · y χB1 (0) dµx (y )
Rd
and
Z
min{|y |1+γ , 1} dµx (y ) < ∞.
A new result
Theorem (Guillen-Schwab, arXiv two days ago!)
Furthermore if I : C 0,γ (Rd ) → C (Rd ) is Lipschitz and satisfies the GCP,
then
Z
L(u, x) =
Rd
u(x + y ) − u(x) − ∇u(x) · y χB1 (0) dµx (y )
and
Z
min{|y |γ , 1} dµx (y ) < ∞.
The importance of min-max formulas
For local elliptic equations
F (D 2 u, ∇u, u, x) = 0
we have that the right hand side can be represented as
min max{fij (x) + cij (x)u(x) + ∇u · bij (x) + tr(Aij (x)D 2 u(x))}
i
j
In this setting, min-max formulas have been of great use, as they allow
us to represent solutions to a PDE as the value functions of zero-sum
differential games, i.e.
Fully nonlinear
Isaacs equation
⇔
elliptic equation
for some game
The importance of min-max formulas
Tools + Problems
in PDEs
⇔
Tools + Problems
in Differential Games
Evans (1984), Souganidis (1985), and Katsoulakis (1995) have used such
representions to derive regularity estimates crucial to existence and
uniqueness for viscosity solutions of first and second order
Hamilton-Jacobi-Bellman equations.
The importance of min-max formulas
Min-max formulas are of great use, as they allow us to represent
solutions to a PDE to the value function of a zero-sum differential game.
Tools + Problems
in PDEs
⇔
Tools + Problems
in Differential Games
They have also played a crucial role in work of Kuo-Trudinger (1992) on
numerical schemes for uniformly elliptic (fully nonlinear) second order
equations.
The importance of min-max formulas
The original motivation for us was twofold:
1) Resolving a standing issue in the integro-differential literature: what is
a reasonsably wide common framework for nonlocal equations? Many
existing results assume the operator is given as a min-max– can we
justify it?
Recent years have seen a great deal of progress in the
existence/uniqueness theory: Barles-Imbert, Jakobsen-Karlsen,
Barles-Chasseigne-Imbert, Mou-Swiech. . .
2) Showing that nonlinear Neumann problems can be analyzed via
integro-differential methods, e.g. homogenization for nonlinear Neumann
problems (Guillen-Schwab 2015).
Ideas of the proof : Incorrect intuition, norm is a max
kxkV =
max
Λ∈V ∗ , kΛk≤1
(|Λ(x)|)
• f : Rd → R, Lipschitz (Touching above by cones)
f (x) − f (y ) ≤ C |x − y | =⇒ f (x) = min (f (y ) + |x − y |)
y ∈Rd
f (x) = min
max
y ∈Rd Λ∈(Rd )∗ ,kΛk≤1
= min
max
y ∈Rd p∈Rd ,kpk≤1
(f (y ) + Λ(x − y ))
(f (y ) + p · (x − y ))
GREAT! y 7→ p · y is linear with the GCP.
Ideas of the proof : Incorrect intuition, norm is a max
kxkV =
max
Λ∈V ∗ , kΛk≤1
(|Λ(x)|)
• I : C 2 (Rd ) → C (Rd ) Lipschitz (Touching by cones)
I (u, x) − I (v , x) ≤ C ku − v k =⇒ I (x) = min (I (v , x) + ku − v k)
v ∈C 2
I (u, x) = min
max
v ∈C 2 Λ∈(C 2 )∗ ,kΛk≤1
(I (v , x) + Λ(u − v ))
=?????
OOOPS! Why can we limit ourselves to only those Λ that also have the
GCP?!?!?!
Ideas of the proof : Assume too much – Fréchet diff.
Lemma
If I has the GCP and I is Fréchet differentiable at u, then DIu also has
the GCP.
Suppose that φ ≤ 0 everywhere with φ(x0 ) = 0.
Then, for any s ≥ 0
u touches u + sφ from above at x0
⇒ I (u + sφ, x0 ) ≤ I (u, x0 ) ∀ s ≥ 0
I (u + sφ, x0 ) − I (u, x0 )
=⇒ DIu (φ, x0 ) = lim
≤ 0.
s→0
s
Ideas of the proof : Assume too much – Fréchet diff.
Assume I is Fréchet differentiable, then
Z 1
I (u, x) − I (v , x) =
(DI )tu+(1−t)v (u − v , x) dt
0
= L(u − v , x), ∀ u, v ∈ C 2
Given u and v , the linear operator
Z 1
L=
(DI )tu+(1−t)v dt
0
lies in the convex hull of the image of f → (DI )f , i.e.
L ∈ D := hull{L | L = (DI )u , for some u ∈ C 2 }
Ideas of the proof : Assume too much – Fréchet diff.
Then, for every v ∈ C 2 define
Kv (u, x) := max{I (v , x) + L(u − v , x)}
L∈D
The key fact is that (from the previous calculation)
I (u, x) ≤ Kv (u, x), ∀ u, v ∈ C 2 , x ∈ M
But the above is an equality when v = u, so
I (u, x) = min max{fvL (x) + L(u, x)}
v ∈C 2 L∈D
where fvL (x) := I (v , x) − L(v , x) ∈ C 0 .
Ideas of the proof : I is only Lipschitz
Operators which are only Lipschitz are very important (e.g. extremal
operators)
Problem: If dim=∞, Lipschitz maps are not necessarily Fréchet
differentiable in a dense set. So, essentially D = ∅.
Strategy:
-
Prove analogue theorem for finite graphs.
Approximate manifold M by a sequence of finite graphs.
Project the operator I to the graph, preserving GCP property.
Obtain min-max formula in the graph. Pass to the limit.
First, let us prove the finite graph version of the theorem.
Ideas of the proof : finite dimensional case
Let X be a finite set, let C (X ) denote the space of all real valued
functions in X and
L : C (X ) → C (X )
Lemma
Assume L : C (X ) → C (X ) is linear. Then ∃ K (x, y ) ≥ 0 defined for
x 6= y and c(x) ≤ 0 such that
X
L(u, x) = c(x)u(x) +
(u(y ) − u(x))K (x, y )
y ∈X \{x}
Ideas of the proof : finite dimensional case
Proof of the lemma
For any y ∈ X , define
ey (x) :=
1 x = y,
0 x=
6 y.
Clearly,
u(x) =
X
u(y )ey (x) ∀ u
y ∈X
L is linear, so
L(u, x) =
X
u(y )L(ey (·), x)
y ∈X
This may be conveniently rearranged as
X
X
L(u, x) = u(x)
L(ey (·), x) +
(u(y ) − u(x))L(ey (·), x)
y ∈X
y ∈X
Ideas of the proof : finite dimensional case
This can be rewritten as
L(u, x) = c(x)u(x) +
X
(u(y ) − u(x)) K (x, y )
y ∈X \{x}
where
K (x, y ) := L(ey (·), x), ∀ x, y ∈ X , x 6= y ,
c(x) := L(1, x), ∀ x ∈ X .
thus K (x, y ) ≥ 0 and c(x) ≤ 0.
The Clarke Subdifferential
Recall I : C (G ) → C (G ) is Lipschitz.
Given f ∈ C (G ), Clarke “subdifferential” of I at f : (DI )f
n
o
(DI )f := hull L | ∃ {fn } s.t. fn → f , L = lim Ln , Ln = (DI )fn
n
The total subdifferential

DI := hull 

[
(DI )f 
f ∈C (G )
Important: Rademacher’s theorem: (DI )f 6= ∅ ∀f ∈ C (Gn ).
The Clarke Subdifferential
The following is a useful property of the Clarke subdifferential.
Lemma
Let I : C (G ) → C (G ), Lipschitz.
Given f , g ∈ C (G ), ∃ L ∈ hull(DI ) such that
I (f ) = I (g ) + L(f − g )
Ideas of the proof : finite dimensional case
We can now conclude just as in the Fréchet case:
I (f , x) = min Kg (f , x)
g ∈C (G )
= min
max {I (g , x) + L(f − g , x)}
g ∈C (G ) L∈hull(DI )
Now, we use the GCP property of I :
hull(DI ) = convex closure of the limits of {Ln }, with Ln = (DI )fn , all of
which have the GCP!
It follows that every L ∈ hull(DI ) has the GCP, and the finite graph
theorem is proved.
Ideas of the proof : approximation and back again
Disclaimer! This is a gross simplification
For each n ∈ N, we have
Gn := [−2n , 2n ] ∩ (2−n Zd )
Gn ⊂ Gn+1 , ( lim Gn ) = Rd
n→∞
Stage
d
I : C 2 (Rd ) → C R Lipschitz
In : C (Gn ) → C (Gn )
In has min-max
DIn
I min-max
“DI = limn→∞ DIn ”
key tool
Whitney Extension + Restriction
Tn I (En2 u)
proved above
proved above
“kIn (u) − I (u)kC (Rd ) → 0” u ∈ Cc3 (Rd )
stability of min-max
What you end up with
Min-Max over discrete operators
In (u, x) = min max (In (v , x) + Ln (u − v , x))
v ∈Cbβ Ln ∈DIn
where
n
Z
∀x ∈ Gn , Ln (u, x) = C (x)u(x) +
u(y ) − u(x) µnx (dy ).
M\{x}
or
Ln (u, x) = Tr(Aδ,n (x)D 2 u(x)) + B δ,n (x) · ∇u(x) + C n (x)u(x)
Z
+
u(y ) − Txδ,β (u, y ) µnx (dy );
M\{x}
WARNING!
The measures µnx from the previous slide are SIGNED MEASURES
The D-to-N
(
F (Uφ , X ) = 0
Uφ = φ
in Ω
on ∂Ω,
Assume F is good enough that weak solutions satisfy regularity
φ ∈ C 1,γ (∂Ω) =⇒ Uφ ∈ C 1,γ (Ω),
and comparison
Uφ |∂Ω = φ ≤ ψ = Uψ |∂Ω =⇒ Uφ ≤ Uψ in Ω.
Defines the D-to-N
I : C 1,γ (∂Ω) → C γ (∂Ω)
I(φ, x) = ∂n Uφ (x),
The D-to-N
• F (U, X ) = −div(A∇U)(X ), A ∈ C γ symmetric, uniformly elliptic
• F (U, X ) = Tr(A(X )D 2 U) + B(X ) · ∇U + C (X )U, A ∈ C γ
symmetric, uniformly elliptic
• F (U, x) = F (D 2 U, ∇U, x), uniformly elliptic,
|F (P, x) − F (P, y )| ≤ C |x − y |γ (1 + kPk)
The D-to-N
Theorem (Bony-Courrège,I as above, for both linear cases of F ,)
I(u, x) =C (x)u(x) + B(x) · ∇τ u(x)
Z
d(x, y )2
+
)dµab
u(y ) − u(x) − χBr0 (x) (∇τ u(x), ∇τ
x (y )
2
∂Ω\{x}
Theorem (Guillen-Schwab arXiv two days ago!, I as above, F
nonlinear)
n
I(u, x) = min max C ab (x)u(x) + B ab (x) · ∇τ u(x)
a
b
Z
d(x, y )2
+
u(y ) − u(x) − χBr0 (x) (∇τ u(x), ∇τ
)dµab
x (y ) }
2
∂Ω\{x}
The D-to-N
Theorem (Guillen-Schwab arXiv two days ago!... continued)
Furthermore,
1. For every γ > 1 we have
Z
sup
min{1, d(x, y )γ } dµab
x (y ) < ∞
x∈∂Ω ∂Ω\{x}
2. There are constants r0 , c, and C such that for all r ∈ (0, r0 )
Z
−1
cr −1 ≤
dµab
∀ x ∈ ∂Ω, r > 0.
x (y ) ≤ Cr
B2r (x)\Br (x)
For example... (Schwab-Silvestre 2016) Hölder regularity when:
Z
•
K (h) dh ≤ (2 − α)Λr −α
B2r \Br
• For every r > 0, there exists a set Ar such that
• Ar ⊂ B2r \ Br .
• Ar is symmetric in the sense that Ar = −Ar .
• |Ar | ≥ µ|B2r \ Br |.
• K (h) ≥ (2 − α)λr −d−α in Ar .
The D-to-N : reverse the flow!
OLD Flow: integro-differential problem on ∂Ω → second order extension
→ send info to ∂Ω
NEW Flow: second order problem in Ω → reduction to int-diff on ∂Ω →
sind info back to problem in Ω
Good motivation: Homogenization
The D-to-N
Fully nonlinear equations / nonlinear boundary operators
Interesting questions
A small sample of many open problems/research directions
1. For what fully nonlinear operators can we prove that the Lévy
measures in the min-max formula fall in the Bass-Levin class?
2. Characterize which nonlocal operators arise as D-to-N maps.
3. What if M = Rd and I is translation invariant? Can the operators in
the min-max be taken to be translation invariant as well?.
4. Extend theorem to singular/unbounded operators such as the
infinity Laplace and p-Laplace.
5. Obtain bounds on the Lévy kernels k(x, y ) when the diffusion
coefficients of L degenerate at ∂D. This is related to questions of
unique continuation (currently being investigated by M. Boratko at
UMass)
The End
Thanks!