A Rigidity Result for Overdetermined Elliptic Problems in the Plane

Ros, Antonio; Ruiz, David; Sicbaldi, Pieralberto

doi:10.1002/cpa.21696

Cited by 33 publications

(20 citation statements)

References 35 publications

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“…Second, Ros-Ruiz-Sicbaldi [31] proved that the BCN conjecture is true in dimension 2 for unbounded domains whose complement is unbounded, such domain must be a half-space. Also, Ros-Ruiz-Sicbaldi [31] constructed exteriors domains different from the exterior of a geodesic ball in R 2 for particular choices of the Lipschitz function f , this gives a counterexample to the BCN conjecture in R 2 in all its generality. Hence, combining the works of Pucci-Serrin, Reichel and Ros-Ruiz-Sicbaldi we have Theorem [27,29,31].…”

Section: Introductionmentioning

confidence: 99%

“…Also, Ros-Ruiz-Sicbaldi [31] constructed exteriors domains different from the exterior of a geodesic ball in R 2 for particular choices of the Lipschitz function f , this gives a counterexample to the BCN conjecture in R 2 in all its generality. Hence, combining the works of Pucci-Serrin, Reichel and Ros-Ruiz-Sicbaldi we have Theorem [27,29,31]. Let f be a non-increasing Lipschitz function and Ω ⊂ R 2 a C 2,α connected domain whose complement is connected.…”

Section: Introductionmentioning

confidence: 99%

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Characterization of f-extremal disks

Espinar

Mazet

2019

Journal of Differential Equations

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We show uniqueness for overdetermined elliptic problems defined on topological disks Ω with C 2 boundary, i.e., positive solutions u to ∆u + f (u) = 0 in Ω ⊂ (M 2 , g) so that u = 0 and ∂ u ∂ η = cte along ∂ Ω, η the unit outward normal along ∂ Ω under the assumption of the existence of a candidate family. To do so, we adapt the Gálvez-Mira generalized Hopf-type Theorem [19] to the realm of overdetermined elliptic problem.Whenfor any x ∈ R * + , we construct such candidate family considering rotationally symmetric solutions. This proves the Berestycki-Caffarelli-Nirenberg conjecture in S 2 for this choice of f . More precisely, this shows that if u is a positive solution to ∆u + f (u) = 0 on a topological disk Ω ⊂ S 2 with C 2 boundary so that u = 0 and ∂ u ∂ η = cte along ∂ Ω, then Ω must be a geodesic disk and u is rotationally symmetric. In particular, this gives a positive answer to the Schiffer conjecture D (cf. [33,35]) for the first Dirichlet eigenvalue and classifies simply-connected harmonic domains (cf. [28], also called Serrin Problem) in S 2 . MSC 2010: 35Nxx; 53Cxx.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Characterization of f-extremal disks

Espinar

Mazet

2019

Journal of Differential Equations

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“…In the harmonic case f = 0 a complete classification of solutions to the problem (1.1) in the plane has been given in [29]. Moreover, the work [21], proves the BCN conjecture in dimension 2 if ∂Ω is connected and unbounded.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 86%

Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

Ruiz¹,

Sicbaldi²,

Wu³

2021

Preprint

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In this paper, we prove the existence of nontrivial unbounded domains Ω ⊂ R n+1 , n ≥ 1, bifurcating from the straight cylinder B × R (where B is the unit ball of R n ), such that the overdetermined elliptic problemhas a positive bounded solution. We will prove such result for a very general class of functions f : [0, +∞) → R. Roughly speaking, we only ask that the Dirichlet problem in B admits a nondegenerate solution. The proof uses a local bifurcation argument.

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“…Partial positive answers to the BCN conjecture in dimension 2 have been given in several works, see [10,13,20,21,31,33]. In [20] the authors show that the conjecture holds in dimension 2 under the hypothesis that ∂Ω is unbounded. The counterexample we give in this paper shows that such hypothesis is actually sharp.…”

Section: Introductionmentioning

confidence: 97%

Solutions to overdetermined elliptic problems in nontrivial exterior domains

2019

Self Cite

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In this paper we construct nontrivial exterior domains Ω ⊂ R N , for all N ≥ 2, such that the problem      −∆u + u − u p = 0, u > 0 in Ω, u = 0 on ∂Ω, ∂u ∂ν = cte on ∂Ω, admits a positive bounded solution. This result gives a negative answer to the Berestycki-Caffarelli-Nirenberg conjecture on overdetermined elliptic problems in dimension 2, the only dimension in which the conjecture was still open. For higher dimensions, different counterexamples have been found in the literature; however, our example is the first one in the form of an exterior domain. 2010 Mathematics Subject Classification. 35J61, 35N25.

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A Rigidity Result for Overdetermined Elliptic Problems in the Plane

Cited by 33 publications

References 35 publications

Characterization of f-extremal disks

Characterization of f-extremal disks

Overdetermined elliptic problems in onduloid-type domains with general nonlinearities

Solutions to overdetermined elliptic problems in nontrivial exterior domains

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