Hamilton—Jacobi Equations and Distance Functions on Riemannian Manifolds

By Carlo Mantegazza, Andrea Mennucci. 2002

233 citations recorded by the source corpus.

Abstract. The paper is concerned with the properties of the distance function from a closed subset of a Riemannian manifold, with particular attention to the set of singularities.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/41078b81f0947390611080723c0e1b0e0e40a296

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In Applied Mathematics

Citations in the corpus

Trade and the Topography of the Spatial Economy (2014, unknown): “Its solution is characterized by the following Eikonal (or Hamilton-Jacobi in the viscosity sense) partial differential equation (see Mantegazza and Mennucci (2003)).”
On Viscosity Solutions of Certain Hamilton–Jacobi Equations: Regularity Results and Generalized Sard's Theorems (2008, methodology): “In [35], Mantegazza and Mennucci present the example of a compact convex set S ∈ R(2) with a C(1,1) boundary for which the set Σ(dS) (where dS denotes the distance function to the set S in R(2)) has positive Lebesgue measure.”
THE DISTANCE FUNCTION FROM THE BOUNDARY IN A MINKOWSKI SPACE (2006, methodology): “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…”
Regularity of optimal transport in curved geometry: The nonfocal case (2010, methodology): “One can show (for instance by pushing the method in [23]) that TCL(x) is included in a countable union of smooth (n − 1)-dimensional Lipschitz manifolds, and this union is locally finite on Σ \ J .”
New formulas for the Laplacian of distance functions and applications (2018, background): “Indeed it is well known that Cp is rectifiable with locally finite codimension one Hausdorff measure (see for instance [43]), but in the literature it seems not to be present any (local) bound on its codimension one Minkowski content.”
Nearly Round Spheres Look Convex (2012, background): “Since then not so much has been found, except for explicit computations in particular geometries (shockingly, the cut locus of the ellipsoid was rigorously described only a few years ago [20]), and local properties, such as the Lipschitz continuity and (n−1)-dimensional rectifiability [6, 21, 25, 30”
MULTI-PHASE STRUCTURAL OPTIMIZATION VIA A LEVEL SET METHOD ∗, ∗∗ (2014, background): “Furthermore, when Ω is C2, Σ has zero Lebesgue measure too [31].”