Sobolev Embeddings for Generalized Ridged Domains

Evans, W. D.; Harris, D.J.

doi:10.1112/plms/s3-54.1.141

Cited by 78 publications

(82 citation statements)

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“…See, for instance, W. D. Evans and D. J. Harris [9], which treats general domains and a slightly different notion of ridge, which is not always closed. In Appendix C we will include a fairly short proof that it is arcwise connected.…”

Section: Introductionmentioning

confidence: 99%

The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton‐Jacobi equations

Nirenberg

2004

Comm Pure Appl Math

106

131

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Section: Introductionmentioning

confidence: 99%

The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton‐Jacobi equations

Nirenberg

2004

Comm Pure Appl Math

106

131

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“…We mention, among other papers, [15] in the framework of general first-order Hamilton-Jacobi equations and Finsler geometry, [14] and [17] in the case of Riemannian manifolds, [10] and [21] for the point of view of nonsmooth analysis in Hilbert spaces, [13] for results applied to the theory of Sobolev spaces, and [9] for applications to causality theory.…”

Section: Introductionmentioning

confidence: 99%

The distance function from the boundary in a Minkowski space

Crasta¹,

Malusa²

2007

Trans. Amer. Math. Soc.

119

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Abstract. Let the space R n be endowed with a Minkowski structure M (that is, M : R n → [0, +∞) is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class C 2 ), and let; y ∈ ∂Ω} be the Minkowski distance of a point x ∈ Ω from the boundary of Ω. We prove that a suitable extension of d Ω to R n (which plays the rôle of a signed Minkowski distance to ∂Ω) is of class C 2 in a tubular neighborhood of ∂Ω, and that d Ω is of class C 2 outside the cut locus of ∂Ω (that is, the closure of the set of points of nondifferentiability of d Ω in Ω). In addition, we prove that the cut locus of ∂Ω has Lebesgue measure zero, and that Ω can be decomposed, up to this set of vanishing measure, into geodesics starting from ∂Ω and going into Ω along the normal direction (with respect to the Minkowski distance). We compute explicitly the Jacobian determinant of the change of variables that associates to every point x ∈ Ω outside the cut locus the pair (p(x), d Ω (x)), where p(x) denotes the (unique) projection of x on ∂Ω, and we apply these techniques to the analysis of PDEs of Monge-Kantorovich type arising from problems in optimal transportation theory and shape optimization.

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“…z n =z then x is the point of intersection of two distance rays, and it follows (see e.g. [EH87,Sect. 3 …”

Section: Remark 2 Functions (4) and (5) Satisfy Conditions (H1)-(h4)mentioning

confidence: 99%

“…If x 2 O lies in the relative interior of a distance ray R x ; then the ray R x is the unique distance ray that contains x; and the set N @O ðxÞ consists of one point, see [EH87,Lemma 3.4]. …”

Section: Remark 2 Functions (4) and (5) Satisfy Conditions (H1)-(h4)mentioning

confidence: 99%