Overdetermined Boundary Value Problems for the  -Laplacian

Buttazzo, Giuseppe; Kawohl, Bernd

doi:10.1093/imrn/rnq071

Cited by 18 publications

(40 citation statements)

References 16 publications

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“…Proposition 9. Assume that the unique viscosity solution to problem (7) or (8) is of class C 1 (Ω), and that In turn, if the unique solution to problem (7) or (8) happens to be a web-function, we can prove that necessarily the cut locus and high ridge of Ω agree. Actually this geometric condition turns out to be necessary and sufficient for the solution being a web-function, according to the result below: Proposition 10.…”

Section: Resultsmentioning

confidence: 99%

“…A viscosity solution to (7) or to (8) is a function u ∈ C(Ω) such that u = 0 on ∂Ω and u is a viscosity solution to the pde −∆ ∞ u = 1 or −∆ N ∞ u = 1, according to the above recalled definitions. The existence and uniqueness of such a viscosity solution has been proved in [29,4] for the Dirichlet problem (7) and in [32,30,31,2] for the Dirichlet problem (8). Concerning regularity, we proved in our previous papers [13] and [11] that, under the assumption that Ω is convex, the unique solution to the above Dirichlet problems is power-concave (precisely, ( 3 4 )-concave in case of problem (7) and ( 1 2 )-concave in case of problem (8)), locally semiconcave, and of class C 1 (Ω).…”

Section: Preliminariesmentioning

confidence: 99%

“…The existence and uniqueness of such a viscosity solution has been proved in [29,4] for the Dirichlet problem (7) and in [32,30,31,2] for the Dirichlet problem (8). Concerning regularity, we proved in our previous papers [13] and [11] that, under the assumption that Ω is convex, the unique solution to the above Dirichlet problems is power-concave (precisely, ( 3 4 )-concave in case of problem (7) and ( 1 2 )-concave in case of problem (8)), locally semiconcave, and of class C 1 (Ω). In case of the Dirichlet problem for the not-normalized operator, such regularity result was established in [13] under the additional assumption that Ω satisfies an interior sphere condition; we are going to remove this restriction in Lemma 13 below, using the fact that an appropriate web function is a supersolution (see Proposition 12).…”

Section: Preliminariesmentioning

confidence: 99%

“…(1) Aim of this paper is to provide a complete characterization of convex domains Ω ⊂ R n where such problems admit a solution. Following the seminal paper by Serrin [34] and the huge amount of literature after it (see for instance [5,6,18,19,20,21,27,35]), overdetermined boundary value problems involving the infinity Laplace operator were firstly considered only few years ago by Buttazzo and Kawohl (see [7]). In fact, due to the high degeneracy of the operator, all the different methods exploited in the literature to obtain symmetry results for overdetermined boundary value problems fail when applied to problems (1)- (2).…”

Section: Introductionmentioning

confidence: 99%

“…In fact, due to the high degeneracy of the operator, all the different methods exploited in the literature to obtain symmetry results for overdetermined boundary value problems fail when applied to problems (1)- (2). In [7], Buttazzo and Kawohl dealt with a simplified version of problems (1)- (2), which consists in looking for solutions having the same level lines as the distance function to the boundary of Ω, which are called web-functions (see Section 2 below). This simplification essentially reduces the problem to a one-dimensional setting, allowing to prove that the existence of a web-solution implies a precise geometric condition on Ω, which is the coincidence of its cut locus and high ridge (see again Section 2 for the definitions).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Characterization of stadium-like domains via boundary value problems for the infinity Laplacian

Crasta¹,

Fragalà

2016

Nonlinear Analysis

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We give a complete characterization, as "stadium-like domains", of convex subsets Ω of R n where a solution exists to Serrin-type overdetermined boundary value problems in which the operator is either the infinity Laplacian or its normalized version. In case of the not-normalized operator, our results extend those obtained in a previous work, where the problem was solved under some geometrical restrictions on Ω. In case of the normalized operator, we also show that stadium-like domains are precisely the unique convex sets in R n where the solution to a Dirichlet problem is of class C 1,1 (Ω).

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Section: Resultsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Characterization of stadium-like domains via boundary value problems for the infinity Laplacian

Crasta¹,

Fragalà

2016

Nonlinear Analysis

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A Viscosity Equation for Minimizers of a Class of Very Degenerate Elliptic Functionals

Ciraolo

2013

Geometric Properties for Parabolic and Elliptic PDE's

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The study of geometric properties of partial differential equations has always attracted the interest of researchers and is now a broad and well-established research area, with contributions that often come from experts from disparate areas of mathematics, such as differential and convex geometry, functional analysis, calculus of variations, mathematical physics, to name a few.The interplay between partial differential equations and geometry has two main aspects: on the one hand, the former is classically a powerful tool for the investigation of important problems coming from differential geometry and, on the other hand, the latter gives useful and often decisive insights in the study of PDE's. Now that basic questions about PDE's, such as existence, uniqueness, stability and regularity of solutions for initial/boundary value problems, have been fairly understood, research on topological and/or geometric properties of their solutions have become more vigorous.Research on geometric aspects for parabolic and elliptic PDE's provides a vast variety of possibilities. Issues currently and actively studied comprehend among others: positivity of solutions; critical points: their structure, possible occurrence and evolution; spike-shaped solutions; symmetry and non-symmetry for ground states and overdetermined boundary value problems; stability of symmetric configurations; convexity, quasi-convexity or starshape of level sets; estimates on geometrically or physically relevant quantities such as surface area and curvature of level surfaces or torsional creep, eigenvalues and eigenfunctions; impact of curvature of the domain on the relevant solutions and their possible behavior for large or short times, and so on. Similarly wide is the assortment of mathematical tools and techniques, analytic and geometric, employed to analyze such issues: functional inequalities such as isoperimetric, Hardy or Brunn-Minkowski inequalities; Pohozaev-type identities; maximum principles; Harnack inequalities; asymptotics for solutions; moving-planes or sliding methods; Bernstein and Liouville-type theorems, viscosity-solutions techniques, et cetera.This volume aims to promote scientific collaboration in this very active area of research, by presenting recent results and informative surveys and by exploring new trends and techniques. It contains original papers and a few survey articles.

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On the Dirichlet and Serrin Problems for the Inhomogeneous Infinity Laplacian in Convex Domains: Regularity and Geometric Results

Crasta¹,

Fragalà

2015

Arch Rational Mech Anal

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Abstract.Given an open bounded subset Ω of R n , which is convex and satisfies an interior sphere condition, we consider the pde −∆∞u = 1 in Ω, subject to the homogeneous boundary condition u = 0 on ∂Ω. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class C 1 (Ω). We then investigate the overdetermined Serrin-type problem, formerly considered in [10], obtained by adding the extra boundary condition |∇u| = a on ∂Ω; by using a suitable P -function we prove that, if Ω satisfies the same assumptions as above and in addition contains a ball with touches ∂Ω at two diametral points, then the existence of a solution to this Serrin-type problem implies that necessarily the cut locus and the high ridge of Ω coincide. In turn, in dimension n = 2, this entails that Ω must be a stadium-like domain, and in particular it must be a ball in case its boundary is of class C 2 .

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Overdetermined Boundary Value Problems for the -Laplacian

Cited by 18 publications

References 16 publications

Characterization of stadium-like domains via boundary value problems for the infinity Laplacian

Characterization of stadium-like domains via boundary value problems for the infinity Laplacian

A Viscosity Equation for Minimizers of a Class of Very Degenerate Elliptic Functionals

On the Dirichlet and Serrin Problems for the Inhomogeneous Infinity Laplacian in Convex Domains: Regularity and Geometric Results

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