Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data

Paul Sacks; Mahamadi Warma

doi:10.3934/dcds.2014.34.761

Article Contents

2014, Volume 34, Issue 2: 761-787. Doi: 10.3934/dcds.2014.34.761

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Semi-linear elliptic and elliptic-parabolic equations with Wentzell boundary conditions and $L^1$-data

Paul Sacks^1, and
Mahamadi Warma^2,

1.
Iowa State University, Department of Mathematics, 396 Carver Hall, Ames, IA 50011
2.
University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377

Received: January 2013

Revised: May 2013

Published: February 2014

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Abstract

Let $Ω\subset\mathbb{R}^N$ ($N\ge 2$) be a bounded domain with a boundary $∂Ω$ of class $C^2$ and let $\alpha,\beta$ be maximal monotone graphs in $\mathbb{R}^2$ satisfying $\alpha(0)\cap\beta(0)\ni 0$. Given $f\in L^1(Ω)$ and $g\in L^1(∂Ω)$, we characterize the existence and uniqueness of weak solutions to the semi-linear elliptic equation $-\Delta u+\alpha(u)\ni f$ in $Ω$ with the nonlinear general Wentzell boundary conditions $-\Delta_{\Gamma} u+\frac{\partial u}{\partial\nu}+\beta(u)\ni g$ on $∂Ω$. We also show the well-posedness of the associated parabolic problem on the Banach space $L^1(Ω)\times L^1(∂Ω)$.

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Mathematics Subject Classification: 35J60, 35J65, 35D05.

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