Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains

Mahamadi Warma

doi:10.3934/cpaa.2013.12.1881

Article Contents

2013, Volume 12, Issue 5: 1881-1905. Doi: 10.3934/cpaa.2013.12.1881

This issue Previous Article Bounded and unbounded oscillating solutions to a parabolic-elliptic system in two dimensional space Next Article Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations

Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains

Mahamadi Warma^1,

1.
University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377

Received: April 2011

Revised: September 2012

Published: September 2013

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Abstract

We show that on a bounded domain $\Omega\subset R^N$ with Lipschitz continuous boundary $\partial \Omega$, weak solutions of the elliptic equation $\lambda u-Au=f$ in $\Omega$ with the boundary conditions $-\gamma\Delta_\Gamma u+\partial_\nu^a u+\beta u=g$ on $\partial \Omega$ are globally Hölder continuous on $\bar \Omega$. Here $A$ is a uniformly elliptic operator in divergence form with bounded measurable coefficients, $\Delta_\Gamma$ is the Laplace-Beltrami operator on $\partial \Omega$, $\partial_\nu^a u$ denotes the conormal derivative of $u$, $\lambda,\gamma>0$ are real numbers and $\beta$ is a bounded measurable function on $\partial Omega$. We also obtain that a realization of the operator $A$ in $C(\bar \Omega)$ with the general Wentzell boundary conditions $(Au)|_{\partial \Omega}-\gamma\Delta_\Gamma u+\partial_\nu^a u+\beta u=g$ on $\partial \Omega$ generates a strongly continuous compact semigroup. Some analyticity results of the semigroup are also discussed.

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Mathematics Subject Classification: Primary: 35J20, 35B65, 35B45,35K05; Secondary: 47D06.

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