## Advances in Differential Equations

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

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

Markus Biegert and Mahamadi Warma

#### 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