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September/October 2010 Some quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on "bad" domains
Markus Biegert, Mahamadi Warma
Adv. Differential Equations 15(9/10): 893-924 (September/October 2010).

## Abstract

Let $p\in[2,N)$, ${\Omega}\subset{\mathbb{R}}^N$ an open set and let $\mu$ be a Borel measure on ${\partial\Omega}$. Under some assumptions on ${\Omega},\mu,f,g$ and $\beta$, we show that the quasi-linear elliptic equation with nonlinear inhomogeneous Robin-type boundary conditions $\begin{cases} -\Delta_pu+c(x)|u|^{p-2}u=f \;& \text{ in }{\Omega} \\ d{\operatorname{\mathsf N}}_p(u) + \beta(x,u)d\mu =gd\mu \;& \text{ on }{\partial\Omega} \end{cases}$ has a unique weak solution which is globally bounded on ${\overline{\Omega}};$ that is, the weak solution $u$ is in $L^\infty({\Omega})$ and its trace $u|_{{\partial\Omega}}$ belongs to $L^\infty({\partial\Omega},\mu)$. Here ${\operatorname{\mathsf N}}_p(u)$ is a generalization of the normal derivative for bad domains. When ${\Omega}$ and $u$ are smooth, then $d{\operatorname{\mathsf N}}_p(u)= | \nabla u | ^{p-2} (\partial u/\partial\nu) d\sigma$ where $\sigma$ is the surface measure and $\nu$ the outer normal to ${\partial\Omega}$. A priori estimates for solutions are also obtained.

## Citation

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Markus Biegert. Mahamadi Warma. "Some quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on "bad" domains." Adv. Differential Equations 15 (9/10) 893 - 924, September/October 2010.

## Information

Published: September/October 2010
First available in Project Euclid: 18 December 2012

zbMATH: 1203.35109
MathSciNet: MR2677423

Subjects:
Primary: 35B45, 35D10, 35J60, 35J65

Rights: Copyright © 2010 Khayyam Publishing, Inc.

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