Dirichlet problem of heat equation for \(C^ 2\) domains
type: Item
instance of: scholarly article
title: Dirichlet problem of heat equation for \(C^ 2\) domains
zbMATH Open document ID: 0678.35042
DOI: 10.1016/0022-0396(89)90094-6
published in: Journal of Differential Equations
publication date: 1989-01-01T00:00:00Z
review text: Let \(t=\tau (x)\) be a \(C^ 2\) function on \({\mathbb{R}}\), \(\Omega\) the region \(\{(x,t):\quad t>\tau (x)\}.\) For example the following theorem is proved: Let \(\sup | \tau ''| \equiv A<+\infty\), \(f\in L_ p\), \(1\leq p<\infty\), and \[ u(x,t)=\int_{{\mathbb{R}}}K(x,t;y,\tau (y))f(y)dy,\quad for\quad (x,t)\in \Omega, \] where \[ K(x,t;y,\tau (y))=\frac{1}{2\sqrt{\pi (t-\tau (y))}}e^{-(x-y)^ 2/4(t-\tau (y))}[1- \tau '(y)\frac{x-y}{t-\tau (y)}]\quad for\quad t>\tau (y);\quad 0\quad for\quad t\leq \tau (y). \] Then u(x,t) is a solution of the heat equation in \(\Omega\), such that at almost every boundary point \((x_ 0,\tau (x_ 0))\) the non-tangential limit \(u^*(x_ 0,\tau (x_ 0))\) exists and \[ u^*(x_ 0,\tau (x_ 0))=f(x_ 0)+\int_{{\mathbb{R}}}K(x_ 0,\tau (x_ 0);y,\tau (y))f(y)dy. \]
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 4111175
zbMATH Keywords: heat equation
zbMATH Keywords: boundary point
zbMATH Keywords: non-tangential limit
author: Robert Kaufman
author: Jang-Mei Wu
reviewed by: Kurt Hawlitschek
MaRDI profile type: Publication
full work available at URL: 0022-0396(89)90094-6
OpenAlex ID: W2024351127
cites work: On functions of bounded mean oscillation
cites work: Parabolic potential theory
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cites work: On heat capacity and parabolic measure
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description: scientific article; zbMATH DE number 4111175