On the characterization of some classes of proximally smooth sets

Crasta, Graziano; Fragalà, Ilaria

doi:10.1051/cocv/2015022

Cited by 17 publications

(24 citation statements)

References 39 publications

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“…It is written in that the result (D) “gives a complete characterization of connected sets of positive reach with empty interior in the plane”, however it is not true for unbounded sets, see Example below. In Section we generalize (D) giving a complete characterization of one‐dimensional sets

A \subset R^{d}

with positive reach.…”

Section: Introductionmentioning

confidence: 99%

On the structure of sets with positive reach

Rataj

Zaj́ıček

2017

Mathematische Nachrichten

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We give a complete characterization of compact sets with positive reach (=proximally $C^1$ sets) in the plane and of one-dimensional sets with positive reach in ${\mathbb R}^d$. Further, we prove that if $\emptyset \neq A\subset{\mathbb R}^d$ is a set of positive reach of topological dimension $0< k \leq d$, then $A$ has its "$k$-dimensional regular part" $\emptyset \neq R \subset A$ which is a $k$-dimensional "uniform" $C^{1,1}$ manifold open in $A$ and $A\setminus R$ can be locally covered by finitely many $(k-1)$-dimensional DC surfaces. We also show that if $A \subset {\mathbb R}^d$ has positive reach, then $\partial A$ can be locally covered by finitely many semiconcave hypersurfaces

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A \subset R^{d}

with positive reach.…”

Section: Introductionmentioning

confidence: 99%

On the structure of sets with positive reach

Rataj

Zaj́ıček

2017

Mathematische Nachrichten

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“…The choice of the above P -functions is due to the fact that their constancy on the whole Ω, if satisfied, gives the crucial information that u and u N are web-functions, and more precisely that they agree with the functions φ Ω and φ N Ω introduced in (11)- (12). We have indeed:…”

Section: Resultsmentioning

confidence: 93%

“…As a companion result, which will be obtained as a consequence of Theorem 3, we establish that being a stadium-like domain is also a necessary and sufficient condition on a convex set Ω for the C 1,1 regularity of the unique solution to the Dirichlet problem (8): Remark 5. By combining Theorems 2, 3 and 4 with Theorem 6 in [12], we infer that, in dimension n = 2, domains Ω where any of the overdetermined problems (1) or (2) admits a solution (or where the unique solution to problem (8) is of class C 1,1 (Ω)) are geometrically characterized as Ω = {x ∈ R 2 : dist(x, S) < ρ Ω } , being the set S := Σ(Ω) = M(Ω) a line segment (possibly degenerated into a point). If in addition ∂Ω is assumed to be of class C 2 , then Ω is a ball (see [12,Theorem 12]).…”

Section: Resultsmentioning

confidence: 99%

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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“…By Proposition 13, this implies that u = φ Ω (with ρ Ω = a 3 /3), and Σ(Ω) = M(Ω). By combining Theorem 22 with the geometric results we obtained in [17], we can provide some geometric information on the shape of domains where the Serrin-type problem (2) admits a solution, according to Corollary 23 below. We emphasize that symmetry may hold or may fail according in particular to the boundary regularity of Ω.…”

Section: Geometric Results For Serrin's Problemmentioning

confidence: 99%

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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On the characterization of some classes of proximally smooth sets

Cited by 17 publications

References 39 publications

On the structure of sets with positive reach

On the structure of sets with positive reach

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

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

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