Symmetrization methods and applications to PDE’s
Vincenzo Ferone
Università di Napoli “Federico II”
CIMPA-UNESCO-EGYPT School
Recent Developments in the Theory of Elliptic PDE
Alexandria, January 26-February 3, 2009
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
1 / 18
Applications of isoperimetric inequality
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
2 / 18
Applications of isoperimetric inequality
Pólya-Szegö inequality
Let u ∈ W 1,p (Rn ) be a non-negative function with compact support, then:
Z
Z
|Du ] |p dx ≤
|Du|p dx.
Rn
V. Ferone (Univ. di Napoli “Federico II”)
Rn
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
2 / 18
Applications of isoperimetric inequality
Pólya-Szegö inequality
Let u ∈ W 1,p (Rn ) be a non-negative function with compact support, then:
Z
Z
|Du ] |p dx ≤
|Du|p dx.
Rn
Rn
If the equality holds, is it true that u = u ] , modulo translations?
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
2 / 18
Applications of isoperimetric inequality
Pólya-Szegö inequality
Let u ∈ W 1,p (Rn ) be a non-negative function with compact support, then:
Z
Z
|Du ] |p dx ≤
|Du|p dx.
Rn
Rn
If the equality holds, is it true that u = u ] , modulo translations?
Yes, if the following condition is assumed:
|{0 < u ] < sup u} ∩ {|Du ] | = 0}| = 0.
[J.E. Brothers - W.P. Ziemer, 1988]
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
2 / 18
First eigenvalue (Lord Rayleigh)
Z
|∇v |2 dx
λ1 (Ω) = min ΩZ
v 2 dx
Ω
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
3 / 18
First eigenvalue (Lord Rayleigh)
Z
Z
2
|∇ψ|2 dx
|∇v | dx
λ1 (Ω) = min ΩZ
=
2
Ω
Z
v dx
Ω
ψ 2 dx
Ω

 −∆ψ = λ1 ψ in Ω

V. Ferone (Univ. di Napoli “Federico II”)
ψ=0
on ∂Ω
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
3 / 18
First eigenvalue (Lord Rayleigh)
Z
Z
2
|∇ψ|2 dx
|∇v | dx
λ1 (Ω) = min ΩZ
=
2
Ω
Z
v dx
Ω
Z
≥
ψ 2 dx
Ω
|∇ψ ] |2 dx
Ω]
≥ Z
(ψ ] )2 dx
Ω]

 −∆ψ = λ1 ψ in Ω

V. Ferone (Univ. di Napoli “Federico II”)
ψ=0
on ∂Ω
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
3 / 18
First eigenvalue (Lord Rayleigh)
Z
Z
2
|∇ψ|2 dx
|∇v | dx
λ1 (Ω) = min ΩZ
=
2
Ω
Z
v dx
Ω
Z
≥
ψ 2 dx
Ω
|∇ψ ] |2 dx
Ω]
≥ λ1 (Ω] )
≥ Z
] 2
(ψ ) dx
Ω]

 −∆ψ = λ1 ψ in Ω

V. Ferone (Univ. di Napoli “Federico II”)
ψ=0
on ∂Ω
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
3 / 18
Applications of isoperimetric inequality
Comparison results for elliptic equations
Consider the problems (aij ξi ξj ≥ |ξ|2 )

 −(aij uxi )xj = f in E

u=0
V. Ferone (Univ. di Napoli “Federico II”)
on ∂E

 −∆v = f ] in E ]

Symmetrization methods (II)
v =0
on ∂E ]
CIMPA-UNESCO-EGYPT
4 / 18
Applications of isoperimetric inequality
Comparison results for elliptic equations
Consider the problems (aij ξi ξj ≥ |ξ|2 )

 −(aij uxi )xj = f in E

u=0
on ∂E

 −∆v = f ] in E ]

on ∂E ]
v =0
then
u ] (x) ≤ v (x),
in E ] .
The equality case has been characterized.
[H.F. Weinberger, 1962], [V.G. Maz’ja, 1969], [G. Talenti, 1976], [A. Alvino P.-L. Lions - G. Trombetti, 1986], [S. Kesavan, 1988], [V.F. - M.R. Posteraro,
1991]
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
4 / 18
Comparison result
For the sake of simplicity we assume that the data are regular and,
furthermore, f ≥ 0, then u ≥ 0.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
5 / 18
Comparison result
For the sake of simplicity we assume that the data are regular and,
furthermore, f ≥ 0, then u ≥ 0.
Main ingredients of the proof
Integration on the levels of u
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
5 / 18
Comparison result
For the sake of simplicity we assume that the data are regular and,
furthermore, f ≥ 0, then u ≥ 0.
Main ingredients of the proof
Integration on the levels of u
Gauss theorem
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
5 / 18
Comparison result
For the sake of simplicity we assume that the data are regular and,
furthermore, f ≥ 0, then u ≥ 0.
Main ingredients of the proof
Integration on the levels of u
Gauss theorem
Co-area formula
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
5 / 18
Comparison result
For the sake of simplicity we assume that the data are regular and,
furthermore, f ≥ 0, then u ≥ 0.
Main ingredients of the proof
Integration on the levels of u
Gauss theorem
Co-area formula
Isoperimetric inequality
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
5 / 18
Comparison result
For the sake of simplicity we assume that the data are regular and,
furthermore, f ≥ 0, then u ≥ 0.
Main ingredients of the proof
Integration on the levels of u
Gauss theorem
Co-area formula
Isoperimetric inequality
Rearrangements properties
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
5 / 18
Comparison result
Z
Z
f
−(aij uxj )xi =
=
u>t
V. Ferone (Univ. di Napoli “Federico II”)
u>t
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
6 / 18
Comparison result
Z
Z
f
−(aij uxj )xi =
=
u>t
(Gauss)
u>t
Z
=
aij uxj
u=t
V. Ferone (Univ. di Napoli “Federico II”)
uxi
≥
|∇u|
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
6 / 18
Comparison result
Z
Z
f
−(aij uxj )xi =
=
u>t
(Gauss)
u>t
Z
=
aij uxj
u=t
uxi
≥
|∇u|
(ellipticity)
Z
≥
|∇u| ≥
u=t
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
6 / 18
Comparison result
Z
Z
f
−(aij uxj )xi =
=
u>t
(Gauss)
u>t
Z
=
aij uxj
u=t
uxi
≥
|∇u|
(ellipticity)
Z
≥
|∇u| ≥
(Cauchy-Schwarz)
u=t
2
Z
1
≥
Z
u=t
u=t
V. Ferone (Univ. di Napoli “Federico II”)
1
|∇u|
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
6 / 18
Comparison result
2
Z
1
Z
f
u>t
≥
Z
u=t
u=t
V. Ferone (Univ. di Napoli “Federico II”)
1 =
|∇u|
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
7 / 18
Comparison result
2
Z
1
Z
f
≥
u>t
Z
u=t
u=t
≥
1/n
nωn µu (t)1−1/n
V. Ferone (Univ. di Napoli “Federico II”)
(co-area + isop. ineq.)
1 =
|∇u|
2
−µ0u (t)
Symmetrization methods (II)
µu (t) = |{u > t}|
CIMPA-UNESCO-EGYPT
7 / 18
Comparison result
2
Z
1
Z
f
≥
u>t
Z
u=t
u=t
≥
(co-area + isop. ineq.)
1 =
|∇u|
1/n
nωn µu (t)1−1/n
2
µu (t) = |{u > t}|
−µ0u (t)
Hardy-Littlewood gives
Z
Z
µu (t)
f ≤
u>t
then
2/n
n2 ωn µu (t)2−2/n
≤
−µ0u (t)
V. Ferone (Univ. di Napoli “Federico II”)
f∗
0
Z
µu (t)
f∗
0
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
7 / 18
Comparison result
∗
0
(−u (s)) ≤
V. Ferone (Univ. di Napoli “Federico II”)
1
2/n
n2 ωn s 2−2/n
Symmetrization methods (II)
Z
s
f∗
0
CIMPA-UNESCO-EGYPT
8 / 18
Comparison result
∗
(−u (s)) ≤
]
∗
n
u (x) = u (ωn |x| ) ≤
Z
1
0
2/n
n2 ωn s 2−2/n
1
2/n
n 2 ωn
s
f∗
0
|E |
1
ωn |x|n
r 2−2/n
Z
r
Z
f ∗ = v (x)
0
where v (x) is the solution of the problem

 −∆v = f ] in E ]

V. Ferone (Univ. di Napoli “Federico II”)
v =0
on ∂E ]
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
8 / 18
Applications of isoperimetric inequality
Analogous comparison results can be proven for several classes of partial
differential equations.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
9 / 18
Applications of isoperimetric inequality
Analogous comparison results can be proven for several classes of partial
differential equations.
Elliptic equations with lower-order terms
−(aij uxi )xj − (bi u)xi + ci uxi + du = f
[A. Alvino - G. Trombetti, 1979], [G. Talenti, 1985], [V.F. - M.R. Posteraro,
1990], . . .
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
9 / 18
Applications of isoperimetric inequality
Analogous comparison results can be proven for several classes of partial
differential equations.
Elliptic equations with lower-order terms
Equations containing p-laplacian operator
−∆p u ≡ −div(|Du|p−2 Du) = f
[G. Talenti, 1979], [C. Maderna - S. Salsa, 1987], [A. Alvino - P.-L. Lions G. Trombetti, 1990], [M.F. Betta - V.F. - A. Mercaldo, 1994], . . .
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
9 / 18
Applications of isoperimetric inequality
Analogous comparison results can be proven for several classes of partial
differential equations.
Elliptic equations with lower-order terms
Equations containing p-laplacian operator
Equations containing the so-called pseudo-p-laplacian operator
e p u ≡ (|ux |p−2 ux )x = div(H(Du)p−1 Hξ (Du)),
∆
i
i
i
i
P
p
1/p
where H(ξ) = ( i |uxi | ) .
[A. Alvino - V.F. -P.-L. Lions - G. Trombetti, 1997], [M. Belloni - V.F. - B.
Kawohl, 2003], . . .
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
9 / 18
Applications of isoperimetric inequality
Analogous comparison results can be proven for several classes of partial
differential equations.
Elliptic equations with lower-order terms
Equations containing p-laplacian operator
Equations containing the so-called pseudo-p-laplacian operator
Hessian type equations like, for example, Monge-Ampère equation
Det(D 2 u) = f .
[G. Talenti, 1981], [K. Tso, 1989], [N.S. Trudinger, 1997], . . .
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
9 / 18
Applications of isoperimetric inequality
Analogous comparison results can be proven for several classes of partial
differential equations.
Elliptic equations with lower-order terms
Equations containing p-laplacian operator
Equations containing the so-called pseudo-p-laplacian operator
Hessian type equations like, for example, Monge-Ampère equation
Problems with Neumann boundary conditions
[C. Maderna - S. Salsa, 1979], [V.F., 1986], [A. Alvino - S. Matarasso - G.
Trombetti, 1994], . . .
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
9 / 18
Applications of isoperimetric inequality
Analogous comparison results can be proven for several classes of partial
differential equations.
Elliptic equations with lower-order terms
Equations containing p-laplacian operator
Equations containing the so-called pseudo-p-laplacian operator
Hessian type equations like, for example, Monge-Ampère equation
Problems with Neumann boundary conditions
Parabolic equations
[C. Bandle, 1976], [J. Mossino - R. Temam, 1981], [J.I. Dı́az - J. Mossino,
1985], [J. Mossino - J.M. Rakotoson, 1986], [A. Alvino - P.-L. Lions - G.
Trombetti, 1990], . . .
...
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
9 / 18
Some consequences of comparison results
Optimal information on the regularity of the solutions, both with
respect to the summability and with respect to continuity properties.
In particular, it is possible to successfully handle the cases where the
data are in spaces with “limit” summability.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
10 / 18
Some consequences of comparison results
Optimal information on the regularity of the solutions, both with
respect to the summability and with respect to continuity properties.
In particular, it is possible to successfully handle the cases where the
data are in spaces with “limit” summability.
For some nonlinear equation the existence of a solution is related to
some condition to be required on the norm of the data. In such cases
the estimates obtained via symmetrization allow to establish optimal
assumptions for the existence.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
10 / 18
Some consequences of comparison results
Optimal information on the regularity of the solutions, both with
respect to the summability and with respect to continuity properties.
In particular, it is possible to successfully handle the cases where the
data are in spaces with “limit” summability.
For some nonlinear equation the existence of a solution is related to
some condition to be required on the norm of the data. In such cases
the estimates obtained via symmetrization allow to establish optimal
assumptions for the existence.
In some cases it has been possible to use symmetrization methods
and the isoperimetric property of the ball in order to prove symmetry
properties for solutions to problems with symmetric data and for
solutions to overdetermined problems.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
10 / 18
Equations with natural growth in the gradient
Let us consider the following problem (aij ξi ξj ≥ |ξ|2 ).

 −(aij uxi )xj = H(x, Du) in Ω

u=0
on ∂Ω
On the term on the right-hand side we make the following assumption
|H(x, ξ)| ≤ |ξ|2 + f (x),
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
f ∈ L∞ (Ω).
CIMPA-UNESCO-EGYPT
11 / 18
Equations with natural growth in the gradient
Let us consider the following problem (aij ξi ξj ≥ |ξ|2 ).

 −(aij uxi )xj = H(x, Du) in Ω

u=0
on ∂Ω
On the term on the right-hand side we make the following assumption
f ∈ L∞ (Ω).
|H(x, ξ)| ≤ |ξ|2 + f (x),
It is possible to obtain a comparison result between the bounded solution
to the given problem and the bounded solution, if it exists, of the following
symmetrized problem:

 −∆v = |Dv |2 + f ] in Ω]

V. Ferone (Univ. di Napoli “Federico II”)
v =0
on ∂Ω]
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
11 / 18
Equations with natural growth in the gradient
Boundary value problems for equations with natural growth in the gradient
have been studied by many authors and the main point in the proof of the
existence of a solution is to give an “a priori” estimate.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
12 / 18
Equations with natural growth in the gradient
Boundary value problems for equations with natural growth in the gradient
have been studied by many authors and the main point in the proof of the
existence of a solution is to give an “a priori” estimate.
Theorem If there exists a bounded solution to the symmetrized problem,
then there exists a solution to the given problem.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
12 / 18
Equations with natural growth in the gradient
Boundary value problems for equations with natural growth in the gradient
have been studied by many authors and the main point in the proof of the
existence of a solution is to give an “a priori” estimate.
Theorem If there exists a bounded solution to the symmetrized problem,
then there exists a solution to the given problem.
[L. Boccardo - F. Murat - J.P. Puel, 1982, . . . ,1992], [J.M. Rakotoson, 1987], [A.
Alvino - P.-L. Lions - G. Trombetti, 1990], [V.F. - F. Murat, 2000], [V.F. - B.
Messano, 2007], . . .
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
12 / 18
Equations with natural growth in the gradient
The existence of the solution to the symmetrized problem can be obtained
under “smallness” assumptions on f .
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
13 / 18
Equations with natural growth in the gradient
The existence of the solution to the symmetrized problem can be obtained
under “smallness” assumptions on f .
Indeed, if v is a bounded solution of problem:

 −∆v = |Dv |2 + f ] in Ω]

v =0
on ∂Ω]
putting w = e v − 1, we have:

 −∆w = wf ] + f ] in Ω]

V. Ferone (Univ. di Napoli “Federico II”)
w =0
on ∂Ω] .
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
13 / 18
A generalized notion of perimeter
Let H : Rn → [0, +∞[ be a convex function satisfying the following
properties:
H(tx) = |t|H(x),
V. Ferone (Univ. di Napoli “Federico II”)
∀x ∈ Rn , ∀t ∈ R.
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
14 / 18
A generalized notion of perimeter
Let H : Rn → [0, +∞[ be a convex function satisfying the following
properties:
H(tx) = |t|H(x),
∀x ∈ Rn , ∀t ∈ R.
α|ξ| ≤ H(ξ) ≤ β|ξ|,
∀ξ ∈ Rn ,
with 0 < α ≤ β.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
14 / 18
A generalized notion of perimeter
Let H : Rn → [0, +∞[ be a convex function satisfying the following
properties:
H(tx) = |t|H(x),
∀x ∈ Rn , ∀t ∈ R.
α|ξ| ≤ H(ξ) ≤ β|ξ|,
∀ξ ∈ Rn ,
with 0 < α ≤ β.
Furthermore, let us suppose that the convex set
K = {x ∈ Rn : H(x) ≤ 1}
has measure |K | equal to ωn .
Sometimes H is called the “gauge” of K .
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
14 / 18
A generalized notion of perimeter
The support function of K
H o (x) = sup < x, ξ >,
ξ∈K
is a convex function such that H and H o are one the polar function of the
other, that is,
H o (x) = sup
ξ6=0
< x, ξ >
H(ξ)
and H(x) = sup
ξ6=0
< x, ξ >
.
H o (ξ)
Clearly H o (x) is the gauge of
K o = {x ∈ Rn : H o (x) ≤ 1}.
We denote by kn the measure of K o .
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
15 / 18
A generalized notion of perimeter
Let E ⊂ Rn be a measuble set of finite measure. It is possible to give a
“generalized” definition of perimeter of E with respect to H such that, for
sufficiently regular domains, it results:
Z
PerH (E ) =
H(ν) dHn−1 (x)
∂E
where ν is the external normal to E .
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
16 / 18
A generalized notion of perimeter
Let E ⊂ Rn be a measuble set of finite measure. It is possible to give a
“generalized” definition of perimeter of E with respect to H such that, for
sufficiently regular domains, it results:
Z
PerH (E ) =
H(ν) dHn−1 (x)
∂E
where ν is the external normal to E .
The following isoperimetric inequality (Wulff Theorem) holds true
1/n
PerH (E ) ≥ nkn |E |1−1/n .
Equality hold if and only if E is a sublevel of H o (Wulff shape), modulo
translations..
[J.E. Taylor, 1979, 1988, . . . , 2002], [I. Fonseca - S. Müller, 1991],
[B. Dacorogna - C.-E. Pfister, 1992], [M. Amar -G. Bellettini, 1994],
[A. Alvino - V. F. - P.-L. Lions - G. Trombetti, 1997]
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
16 / 18
A problem with symmetric data

n
X

∂


−
(H(Du))n−1 Hξi (Du) = f (u) in Ω

∂xi
i=1




u=0
on ∂Ω
where Ω = {x ∈ Rn : H o (x) < R}, with R > 0, n ≥ 2, has the Wulff
shape.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
17 / 18
A problem with symmetric data

n
X

∂


−
(H(Du))n−1 Hξi (Du) = f (u) in Ω

∂xi
i=1




u=0
on ∂Ω
where Ω = {x ∈ Rn : H o (x) < R}, with R > 0, n ≥ 2, has the Wulff
shape.
We suppose that f : R → R is such that f (s) > 0 and f (s) ≤ c1 |s|r + c2
for s > 0 and for some r > 0.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
17 / 18
A problem with symmetric data

n
X

∂


−
(H(Du))n−1 Hξi (Du) = f (u) in Ω

∂xi
i=1




u=0
on ∂Ω
where Ω = {x ∈ Rn : H o (x) < R}, with R > 0, n ≥ 2, has the Wulff
shape.
We suppose that f : R → R is such that f (s) > 0 and f (s) ≤ c1 |s|r + c2
for s > 0 and for some r > 0.
Theorem If H is strictly convex, H(ξ)n is of class C 2 (Rn ) and f satisfies
the above assumptions, then the level sets of any positive solution to the
given problem have the Wulff shape associated to H and then are
homothetic to Ω.
[B. Gidas - W. Ni - L. Nirenberg, 1979], [P.-L. Lions, 1981], [S. Kesavan - F.
Pacella, 1994], [M. Belloni - V.F.- B. Kawohl, 2003]
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
17 / 18
Overdetermined problems
.
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
18 / 18
Overdetermined problems
Let Ω ⊂ Rn an open, bounded, connected set with boundary of class C 2
and let u ∈ C 2 (Ω).
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
18 / 18
Overdetermined problems
Let Ω ⊂ Rn an open, bounded, connected set with boundary of class C 2
and let u ∈ C 2 (Ω).

∆u = n in Ω







u=0
on ∂Ω




∂u



= 1 on ∂Ω
∂νx
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
18 / 18
Overdetermined problems
Let Ω ⊂ Rn an open, bounded, connected set with boundary of class C 2
and let u ∈ C 2 (Ω).

∆u = n in Ω







|x|2 − 1
u=0
on ∂Ω
Ω
is
the
unit
ball,
u
=
⇒
2




∂u


modulo translations

= 1 on ∂Ω
∂νx
[J. Serrin, 1971]
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
18 / 18
Overdetermined problems
Let Ω ⊂ Rn an open, bounded, connected set with boundary of class C 2
and let u ∈ C 2 (Ω).

∆u = n in Ω







|x|2 − 1
u=0
on ∂Ω
Ω
is
the
unit
ball,
u
=
⇒
2




∂u


modulo translations

= 1 on ∂Ω
∂νx
[J. Serrin, 1971]
Above result has been proven with a new approach based also on
symmetrization methods and it has been extended to the case of hessian
operators.
[B. Brandolini - C. Nitsch - P. Salani - C. Trombetti, 2007]
V. Ferone (Univ. di Napoli “Federico II”)
Symmetrization methods (II)
CIMPA-UNESCO-EGYPT
18 / 18