Geometric properties of solutions of Hamilton-Jacobi equations
type: Item
instance of: scholarly article
title: Geometric properties of solutions of Hamilton-Jacobi equations
zbMATH Open document ID: 0535.35005
DOI: 10.1016/0022-0396(85)90005-1
author: Martino Bardi
published in: Journal of Differential Equations
publication date: 1985-01-01T00:00:00Z
review text: The paper studies some geometric properties of the level sets of the solution of the following Cauchy problem for a Hamilton-Jacobi equation: \(u_ t+H(x,t,Du)=0\) in \(R^ n\times(0,\infty)\); \(u(\cdot,0)=u_ 0(\cdot)\) in \(R^ n\), employing the method of moving parallel planes, previously used for elliptic and parabolic PDE's [see e.g. \textit{B. Gidas, W.-M. Ni} and \textit{L. Nirenberg}, Commun. Math. Phys. 68, 209-243 (1979; Zbl 0425.35020)]. The adaptation of this method to Hamilton-Jacobi equations is possible in the framework of the theory of viscosity solutions [see e.g. \textit{M. G. Crandall, L. C. Evans} and \textit{P. L. Lions}, Trans Am. Math. Soc. 282, 487-502 (1984)], using the comparison theorem for such solutions as a kind of maximum principle. Under suitable assumptions, the main of which are the radial symmetry of the Hamiltonian and nonnegative and compact supported initial data, it is shown that at some fixed time to every point of the boundary of a level set it is possible to attach two open truncated cones, one lying inside the level set and the other outside. This implies that the boundary of the level set is a Lipschitz surface. Under the additional hypothesis that the level sets are expanding it is shown that they become asymptotically spherical as \(t\to \infty\), the convergence taking place in the Lipschitz norm. Finally conditions are given under which the level sets are expanding.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 35A30
Mathematics Subject Classification ID: 49L99
zbMATH DE Number: 3849674
zbMATH Keywords: geometric properties
zbMATH Keywords: Hamilton-Jacobi equations
zbMATH Keywords: viscosity solutions
zbMATH Keywords: method of moving parallel planes
zbMATH Keywords: spherical symmetry of solutions
zbMATH Keywords: Cauchy problem
zbMATH Keywords: comparison theorem
MaRDI profile type: Publication
full work available at URL: 0022-0396(85)90005-1
OpenAlex ID: W2010332444
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description: scientific article; zbMATH DE number 3849674