Quasilinear divergence form parabolic equations in Reifenberg flat domains

Dian Palagachev; Lubomira G. Softova

doi:10.3934/dcds.2011.31.1397

Article Contents

2011, Volume 31, Issue 4: 1397-1410. Doi: 10.3934/dcds.2011.31.1397

This issue Previous Article Solving strongly monotone variational and quasi-variational inequalities Next Article Estimates of the derivatives of minimizers of a special class of variational integrals

Quasilinear divergence form parabolic equations in Reifenberg flat domains

Dian Palagachev^1, and
Lubomira G. Softova^2,

1.
Department of Mathematics, Polytechnic University of Bari, 4 E. Orabona Str., 70 125 Bari
2.
Dipartimento di Ingegneria Civile, Seconda Università di Napoli, Via Roma, 29; 81 031 Aversa

Received: January 31, 2010

Revised: August 31, 2010

Published: September 2011

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Abstract

We derive weak solvability and higher integrability of the spatial gradient of solutions to Cauchy--Dirichlet problem for divergence form quasilinear parabolic equations $$ \begin{equation} \left\{\begin{array}{l} u_t-\mathrm{div\,}\big(a^{ij}(x,t,u)D_ju+a^i(x,t,u)\big)=b(x,t,u,Du) &\quad \text{in}\ Q,\\ u=0 &\quad \text{on}\ \partial_p Q, \end{array} \right. \end{equation} $$ where $Q$ is a cylinder in $\mathbb{R}^n\times(0,T)$ with Reifenberg flat base $\Omega.$ The principal coefficients $a^{ij}(x,t,u)$ of the uniformly parabolic operator are supposed to have small $BMO$ norms with respect to $(x,t)$ while the nonlinear terms $a^i(x,t,u)$ and $b(x,t,u,Du)$ support controlled growth conditions.

Keywords:

Divergence form quasilinear parabolic equations,
Weak solvability,
Discontinuous coefficients,
Reifenberg flat domain,
Small $BMO,
$ Controlled growth.

Mathematics Subject Classification: Primary: 35K59; Secondary: 35K61, 35B65, 35R05.

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