Serrin’s type overdetermined problems in convex cones

Ciraolo, Giulio; Roncoroni, Alberto

doi:10.1007/s00526-019-1678-x

Cited by 27 publications

(37 citation statements)

References 40 publications

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“…Let us observe that the hypothesis that the solution u belongs to W 1,∞ (Ω) ∩ W 2,2 (Ω) is automatically satisfied when Γ and Γ 1 intersect orthogonally, as proved in [18]. We also refer the reader to the recent works [6,9,13].…”

Section: Introductionmentioning

confidence: 92%

Isoperimetric cones and minimal solutions of partial overdetermined problems

Pacella¹,

Tralli²

2021

Publicacions Matemàtiques

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In this paper we consider a partial overdetermined mixed boundary value problem in domains inside a cone as in [18]. We show that, in cones having an isoperimetric property, the only domains which admit a solution and which minimize a torsional energy functional are spherical sectors centered at the vertex of the cone. We also show that cones close in the C 1,1 -metric to an isoperimetric one are also isoperimetric, generalizing so a result of [1]. This is achieved by using a characterization of constant mean curvature polar graphs in cones which improves a result of [18].

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Section: Introductionmentioning

confidence: 92%

Isoperimetric cones and minimal solutions of partial overdetermined problems

Pacella¹,

Tralli²

2021

Publicacions Matemàtiques

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“…It appears in condensed-matter and high-energy physics [31,40], and it is naturally studied in quasiconformal geometry (see for instance [29]). The study of symmetry problems in convex cones has recently attracted the interest of many authors, see for instance [6,12,13,22,25,36,37,42]. As far as the authors know, overdetermined capacity problems in convex cones have not been considered so far, even in the Euclidean case.…”

Section: Introductionmentioning

confidence: 99%

“…Regarding the variants for cones, rigidity results of Serrin type were first obtained in Pacella-Tralli [37], where they considered an interior overdetermined problem inside a smooth convex cone and gave a characterization of spherical sectors following the approaches in [5,50]. Then the first author and Roncoroni [13] generalized that to more general elliptic operators which are possibly degenerate as well as to space forms. More generally, during the last decade, much interest has been devoted to other parallel problems in convex cones and anisotropic setting (see for instance [6,12,15,22,42]).…”

Section: Introductionmentioning

confidence: 99%

An exterior overdetermined problem for Finsler $N$-laplacian in convex cones

Ciraolo¹,

Li²

2021

Preprint

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We consider a partially overdetermined problem for anisotropic N -Laplace equations in a convex cone Σ intersected with the exterior of a bounded domain Ω in R N , N ≥ 2. Under a prescribed logarithmic condition at infinity, we prove a rigidity result by showing that the existence of a solution implies that Σ ∩ Ω must be the intersection of the Wulff shape and Σ. Our approach is based on a Pohozaev-type identity and the characterization of minimizers of the anisotropic isoperimetric inequality inside convex cones.

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“…A closely related overdetermined problem in a hemi-sphere has been considered by Qiu and the second author [15] (see also [7]). We also call attention to a similar partially overdetermined problem in a convex cone which has been considered recently by Pacella-Tralli [22] (see also [6] for its generalization to general elliptic operators). Their partially overdetermined BVP is of mixed Dirichlet-Neumann type.…”

Section: Introductionmentioning

confidence: 99%