Advances in Differential Equations

Some quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on "bad" domains

Markus Biegert and Mahamadi Warma

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Adv. Differential Equations Volume 15, Number 9/10 (2010), 893-924.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355854615

Mathematical Reviews number (MathSciNet)
MR2677423

Zentralblatt MATH identifier
1203.35109

Subjects
Primary: 35J60: Nonlinear elliptic equations 35B45: A priori estimates 35D10 35J65: Nonlinear boundary value problems for linear elliptic equations

Citation

Biegert, Markus; Warma, Mahamadi. Some quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on "bad" domains. Adv. Differential Equations 15 (2010), no. 9/10, 893--924. https://projecteuclid.org/euclid.ade/1355854615.


Export citation