Partial regularity for evolution problems with discontinuity
type: Item
instance of: scholarly article
title: Partial regularity for evolution problems with discontinuity
zbMATH Open document ID: 0854.35020
DOI: 10.1007/BF02568295
author: Yoshihiro Tonegawa
author: Robert Hardt
published in: Manuscripta Mathematica
publication date: 1997-01-20T00:00:00Z
full work available at URL: https://eudml.org/doc/156184
review text: For a bounded Lipschitz domain \(\Omega\) in \(\mathbb{R}^n\) and a \(C^1\) convex linear-growth function \(\phi(p)\) on \(\mathbb{R}^n\), the first author and Zhou studied the existence and uniqueness questions on the evolution problem \[ {\partial u\over \partial t}= \sum^n_{i= 1} {\partial\over \partial x_i} \Biggl[ {\partial\phi\over \partial p_i} (\nabla u)\Biggr], \] with suitable Dirichlet boundary data in [the first author and \textit{X. Zhou}, Commun. Partial Differ. Equations 19, 1879-1907 (1994; Zbl 0811.35061)]. The unique weak solutions belong to the function space \(L^2([0, \infty]: \text{BV}(\Omega))\), so that they may in general have discontinuities in the space variable at each time. In fact, depending on the function \(\phi\), the initial and boundary data, some weak solutions do have discontinuities as one can demonstrate with simple examples. In this paper, we establish certain partial regularity results for such solutions in case the space dimension is equal to one or two, with a necessary local coerciveness assumption on \(\phi\).
Mathematics Subject Classification ID: 35D10
Mathematics Subject Classification ID: 35K65
zbMATH DE Number: 885538
zbMATH Keywords: partial regularity
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 885538