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

By Yanyan Li, L. Nirenberg. 2003

144 citations recorded by the source corpus.

We study the distance function to the boundary, Finsler geometry and the singular set of viscosity solutions of some Hamilton-Jacobi equations.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dd930982b07dde8a80557ea79a676e9a898d63f2

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In Applied Mathematics

In Astronomy and Astrophysics

In Geometry and Topology

Citations in the corpus

THE DISTANCE FUNCTION FROM THE BOUNDARY IN A MINKOWSKI SPACE (2006, methodology): “In the same setting, Itoh and Tanaka [14] have proven that, if Ω is of class C ∞ , then the function l is Lipschitz continuous on ∂ Ω. Recently, Li and Nirenberg [15] have reﬁned this result, extending it to the case of Finsler manifolds.”
A geometric characterization of a sharp Hardy inequality (2011, background): “[27] Let Ω ⊂ Rn+1 and S ⊂ Ω be the singular set defined in (2.”
Regularity of the distance function to the boundary (2005, unknown): “2 6”
Generalizations and Properties of the Principal Eigenvalue of Elliptic Operators in Unbounded Domains (2010, unknown): “Since is uniformly of class C (2;1), we know from [21] that, for " > 0 small enough, the distance function d belongs to W (2;1).”
Nearly Round Spheres Look Convex (2012, unknown): “continuity and (n 1)-dimensional rectifiability [6, 21, 25 , 30].”
A symmetry problem in the calculus of variations (2006, background): “It is known that if∂Ω is of classC2,1, thenτ is Lipschitz continuous on ∂Ω (see [18, 17]); for less regular domains the Lipschitz continuity may fail, but continuity is preserved if ∂Ω is of classC2 (see [5, 6]).”
On Viscosity Solutions of Certain Hamilton–Jacobi Equations: Regularity Results and Generalized Sard's Theorems (2008, methodology): “In this spirit, by the classical method of characteristics and under additional assumptions on the data, several authors obtained results on the regularity of u and its singular set, see for instance [ 32 ], [35], [36], [45].”
Nonlinear evolution by mean curvature and isoperimetric inequalities (2006, methodology): “We now want to apply a result of Li and Nirenberg [17] or equivalently Itoh and Tanaka [15], which says: Given a bounded open set S ⊂ Rn+2 with smooth boundary, define G to be the largest open subset of S such that every point x in G has a unique closest point on ∂S.”
Regularity of optimal transport in curved geometry: The nonfocal case (2010, result): “Σ = ⋃ {t} × cut(yt) ⊂ [0, 1] ×M has finite Hn measure in view of the results in [13, 18].”