Convex-monotone functions and generalized solution of parabolic Monge--Ampère equation
type: Item
instance of: scholarly article
title: Convex-monotone functions and generalized solution of parabolic Monge--Ampère equation
zbMATH Open document ID: 1014.35042
DOI: 10.1016/S0022-0396(02)00024-4
author: Li Chen
author: Guanglie Wang
author: Songzhe Lian
published in: Journal of Differential Equations
publication date: 2003-04-02T00:00:00Z
review text: The authors extend and improve their previous results [J. Partial Differ. Equations 14, 149-162 (2001; Zbl 0990.35034)] on the existence of generalized solutions of the first initial-boundary value problem for the parabolic Monge-Ampère equation \[ \begin{aligned} -u_t \det D^2u & = f(x,t) \quad\text{in}\quad Q=\Omega\times(0,T],\\ u & = \varphi(x,t) \quad\text{on}\quad\partial_p Q.\end{aligned}\tag{*} \] The paper contains two main results. The first is the Hölder continuity in \(t\) of \(u\) if \(\varphi(t,x_0)\) is Hölder continuous in \(t\) for each \(x_0\in \partial\Omega\). The second is a geometric characterization of the convex-monotone solution \(U\) of (*) with \(f\equiv 0\) as follows: \[ U(x,t) = \sup \{ l(x): l \text{ is affine and } l(x)\leq\varphi(x,0) \text{ in }\Omega, \] \[ l(x) \leq \varphi(x,t) \text{ on } \partial\Omega \}, \qquad (x,t)\in \bar Q. \] Using these results the authors establish the existence of generalized solutions of (*) under somewhat weaker assumptions than required in previous work.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 1891254
zbMATH Keywords: Hölder continuity in \(t\)
reviewed by: John Urbas
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1891254