International Conference on Geometric Analysis
Geometry and topology of overdeterminedelliptic problems
Pieralberto Sicbaldi
Joint work with Antonio Ros
Domaine de Seillac
September 2012
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The problem:
To understand the geometry of domains Ω ∈ Rn that support a
solution of the over-determined system
∆u+ f(u) = 0 in Ω
u > 0 in Ω
u = 0 on ∂Ω
∂u
∂ν= constant on ∂Ω
Definition: If such problem is solvable, Ω is called an f -extremaldomain.
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Serrin, 1971, Arch. Rat. Mech. An.
If f is Lipschitz and Ω is a C2 bounded domain where thereexists a solution u to the problem
∆u+ f(u) = 0 in Ω
u > 0 in Ω
u = 0 on ∂Ω
∂u
∂ν= constant on ∂Ω
then Ω is a ball.
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The moving plane argument
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Reichel, 1997, Arch. Rat. Mech. An.
If u is a C2 function on Rn\Ω such that
∆u+ f(u) = 0 in Rn\Ω
u = a on ∂Ω
∂u
∂ν= c on ∂Ω
0 ≤ u ≤ a in Rn\Ω
u = 0 at ∞
∣
∣
∣
∣
∣
∣
∣
∣
∣
Ω bounded
Rn\Ω connected
f Lipschitz
f nonincreasing
then Ω is a ball.
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Berestycki, Caffarelli, Nirenberg, 1997, Comm. Pure Appl. Math.
If Ω = xn > ϕ(x1, ..., xn−1), where ϕ is a C2 function verifying
for any τ ∈ Rn−1, uniformly, lim
|x|→∞(ϕ(x+ τ)− ϕ(x)) = 0
and
∆u+ f(u) = 0 in Ω
u = 0 on ∂Ω
∂u
∂ν= c on ∂Ω
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
u bounded
exist 0 < s0 < s1 < µ :
f(s) > 0 on (0, µ)
f(s) ≤ 0 for s ≥ µ
f(s) ≤ δ0s on (0, s0) , δ0 > 0
f nonincreasing on (s1, µ)
then Ω is a half-space.
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Conjecture of Berestycki, Caffarelli and Nirenberg
Communication on Pure and Applied Mathematics, (1997).
∆u+ f(u) = 0 in Ω
u > 0 in Ω
u = 0 on ∂Ω
∂u
∂ν= constant on ∂Ω ,
∣
∣
∣
∣
∣
∣
∣
∣
EXTRA HYPOTHESIS
Rn\Ω connected
u bounded
⇓
Ω is a half space, or a ball, or a cylinder Rj ×B (where B is a ball) orthe complement of one of these three exemples.
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Farina & Valdinoci, 2009, Arch. Rat. Mech. An.
∆u+ f(u) = 0 in Ω
u > 0 in Ω
u = 0 on ∂Ω
∂u
∂ν= constant on ∂Ω ,
∣
∣
∣ f Lipschitz
1. If u ∈ C2(Ω) ∩ L∞(Ω) and f(t) ≥ λt, then Ω cannot be auniformly Lipschitz epigraph.
2. If u ∈ C2(Ω) ∩ L∞(Ω), and [n = 2] or [n = 3 and f(t) ≥ 0],then Ω cannot be a coercive uniformly Lipschitz C3 epigraph.
3. If n = 2 and u ∈ C2(Ω) has bounded gradient and isincreasing in one of the variables, then Ω is a half-plane.
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Theorem (—, 2010, Calc. Var. PDEs)
For n ≥ 3 there exists perturbations Ω of the cylinder Bn−1 × R
(here Bn−1 is the unit ball in Rn−1), with boundary of revolution
and periodic in the direction of the axis of the cylinder, wherethere exists a periodic and positive solution to
∆u+ λu = 0 in Ω
u = 0 on ∂Ω
∂u
∂ν= constant on ∂Ω
x
t
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Schlenk & —, 2012, Adv. Math.
The result is true also in dimension 2. But this is nota counterexemple to the conjecture.
x
t4−4
The complement of such domain is not connected.
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The conjecture in dimension 2
Theorem (Ros & — , 2012)
Suppose there exists λ > 0 such that f(t) ≥ λ t.
With this extra hypothesis, the conjecture ofBerestycki-Caffarelli-Nirenberg in dimension 2 is true.
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A stronger version
Theorem (Ros & — , 2012)
Let us suppose that f is a Lipschitz function such that thereexists λ > 0 such that f(t) ≥ λ t. If Ω a C2-domain of R2 withconnected complement, where one can solve
∆u+ f(u) = 0 in Ω
u > 0 in Ω
u = 0 on ∂Ω
∂u
∂ν= constant on ∂Ω ,
then Ω is a ball. Remark: we do not suppose u bounded.
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The proof of the theorem (1)
Step 1 . ∂Ω has only one connected component.
Then, there are only three possibilities for Ω:
1. Ω is a bounded connected open domain.
2. Ω is the complement of a compact domain.
3. ∂Ω is an open curve that separated R2 in two connected
components, and Ω is one of such components.
Ω
Ω
Ω
By the Serrin’s result, in the first case Ω must be a ball.
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The proof of the theorem (2)
Let Rλ be the radius of balls whose first eigenvalue of theLaplacian with 0 Dirichlet boundary condition is λ.
Step 2. If R ≥ Rλ then Ω does not contain any closed ball ofradius R.
Ω
BRλ(p)
graph of u graph of vǫ
Corollary . Ω can not be the complement of a compact domain.
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Overdetermined problems and CMC surfaces
Theorem (Serrin, 1971) . In Rn the only bounded domains
where the previous overdetermined problem can be solved arethe balls.
Main idea : the moving plane argument, introduced some yearsbefore in the CMC surface theory...
...Theorem (Alexandroff, 1962) . In Rn the only enbedded
compact mean curvature hypersurfaces are the spheres.
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Meeks, 1989, J. Diff. Geom.
Definition. A surface has finite topology if
1. it is a compact surface
2. outside of a big ball, the surface is done of a finite numberof noncompact components diffeomorphic to Sn−1 × R+,called ends.
Theorem . Let S be a properly embedded finite topologynonzero CMC surface in R
3. Then S cannot have only one end.
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The proof of our theorem: (3)
Definition. We say that a domain has finite topology if
1. it is bounded domain, or
2. it is the complement of a compact domain, or
3. outside of a big ball, the domain is done of a finite numberof noncompact components diffeomorphic to Bn−1 × R+,called ends
To prove our main theorem it is enough to prove:
Proposition . Ω cannot have only one end.
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The proof of the proposition (1a)
Let L be an straight line in R2 and let L+ and L− be the two
connected components of R2\L. If Ω ∩ L+ has a boundedconnected component C, then:
C
Ω
L
C ′
Step 1. The closure of ∂C ∩ L+ is a graph on ∂C ∩ L
Step 2. The maximum distance h(C) of ∂C to L satisfies
h(C) ≤ 3Rλ.
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The proof of the proposition (1b)
Step 2. The maximum distance h(C) of ∂C to L satisfies
h(C) ≤ 3Rλ.
bR0−Ra
p q
p′
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The proof of the proposition (2)
Let E be an end of Ω.
Step 3. There exist a line L in R2 such that E stays at bounded
distance from L.
E
L
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The proof of the proposition (3)
Step 4. Ω cannot have only one end.
3Rλ + ǫ
E
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Korevaar, Kusner & Solomon, 1989, J. Diff. Geom.
Let S be a properly embedded finite topology nonzero CMCsurface in R
3 contained in a cylinder. Then S is surface ofrevolution.
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An other result
Theorem (Ros, —, 2012). Let Ω be an f -extremal domain ofR2. If Ω is contained in a half-plane, then Ω is either a ball or
(after a rigid motion) there exists a C2 positive functionϕ : R −→]0,∞[ such that either
i. the domain Ω is an epigraph y > ϕ(x), or
ii. ϕ is bounded and Ω is the symmetric domain |y| < ϕ(x).
Ω
Ω
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On the conjecture of Berestycki-Caffarelli-Nirenberg
Farina & Valdinoci (2009, Arch Rat. Mech. An.) : in dimension2, if u is increasing in one variable and has bounded gradient,then the associated f -extremal domain Ω is a half-plane.
if ∇u is bounded
A straightforward consequence, using our results, is the:
Corollary (Ros, —, 2012) . In dimension 2, if Ω is contained in ahalf-plane the conjecture of Berestycki-Caffarelli-Nirenberg istrue.
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The end
Thank you!
Reference
A. Ros, P. Sicbaldi. Geometry and Topology of someoverdetermined elliptic problems. Preprint (arxiv).
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