The regularity of generalized solutions of Hamilton--Jacobi equations
type: Item
instance of: scholarly article
title: The regularity of generalized solutions of Hamilton--Jacobi equations
zbMATH Open document ID: 1072.35056
author: Xianqiang Yang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-02-22T00:00:00Z
review text: The Cauchy problem for Hamilton-Jacobi equations of the form \[ \begin{gathered} u_t+ H(t,\nabla_x u)= 0,\quad (t,x)\in\Omega= (0,T)\times \mathbb{R}^n,\\ u(0,x)= \sigma(x),\quad x\in \mathbb{R}^n,\end{gathered} \] is considered. By using the method of envelope, Hopf established two well-known formulas for the Lipschitz solutions of the kind (I) \(u(t,x)= \min_{y\in\mathbb{R}^n}\,\{\sigma(y)+ tH^*({x- y\over t})\}\) and (II) \(u(t,x)= \max_{q\in\mathbb{R}^n}\,\{(x, q)- \sigma^*(q)- tH(q)\}\). The author presents some results on the relationship between two Hopf's formulas and characteristics. A generalization of a result of Lions for formula (I) and another one for the formula (II), are established. Then, the domain where the generalized solutions given by Hopf's formulas are differentiable is studied.
reviewed by: Vasile Iftode
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 2136553
zbMATH Keywords: Hopf's formulas
zbMATH Keywords: Strip of differentiability
MaRDI profile type: Publication
full work available at URL: j.na.2004.07.034
OpenAlex ID: W2055820821
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DOI: 10.1016/J.NA.2004.07.034
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description: scientific article; zbMATH DE number 2136553