Parabolic problems with mixed variable lateral conditions: an abstract approach
type: Item
instance of: scholarly article
title: Parabolic problems with mixed variable lateral conditions: an abstract approach
zbMATH Open document ID: 0890.35061
DOI: 10.1016/S0021-7824(97)89955-2
author: Giuseppe Savaré
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1998-06-29T00:00:00Z
review text: The author presents a new approach to initial-boundary value problems for linear parabolic differential equations of second order with mixed and time-dependent boundary conditions. In detail, the following boundary conditions are considered: If \(\Omega\) is a uniformly \(C^{1,1}\) open set of \(\mathbb{R}^N\) with boundary \(\Gamma\) and \(\Gamma^t_0\) is a uniform family of \(C^{1,1}\) submanifolds of \(\Gamma\) \((t\in [0,T])\), then \(u(x,t)= g_0(x, t)\) on \(\Sigma_0:= \bigcup_{t\in(0, T)}\Gamma^t_0\times \{t\}\) and \((\partial u/\partial\nu_A)(x,t)= g_1(x, t)\) on \((\Gamma\times (0,T))\backslash\overline\Sigma_0\). Here \(\nu_A\) is the conormal vector related to the elliptic operator \(A\) in \(\partial_tu+ Au=f\). The author is interested in sufficiently wide conditions on the family \((\Gamma^t_0)_{t\in[0, T]}\) in order to obtain regularity of the type: If \(f\in L^2(\Omega\times (0,T))\) and \(u(x,0)\), \(g_0\), \(g_1\) are in suitable trace spaces, then \(\partial_tu\), \(Au\in L^2(\Omega\times (0,T))\), \(|\nabla u(.,t)|_{L^2(\Omega)}\in L^\infty(0, T)\). He proves a corresponding theorem if \(\sup_{x\in\Gamma^t_0} d(x,\Gamma^s_0)\leq \int^t_s\rho(\xi) d\xi\) for \(s<t\) (with \(\rho\in L^4(0,T)\)). This theorem follows from an abstract result on evolution equations on variable domains. The abstract theory is an essential part of the present paper. An application of the abstract theory to parabolic problems in non-cylindrical domains is also given.
Mathematics Subject Classification ID: 35K20
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35D10
zbMATH DE Number: 1022296
zbMATH Keywords: mixed and time-dependent boundary conditions
zbMATH Keywords: evolution equations on variable domains
zbMATH Keywords: non-cylindrical domains
reviewed by: Wolfgang Watzlawek
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1022296