Some quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on “bad” domains. (English) Zbl 1203.35109

Biegert, Markus; Warma, Mahamadi

Adv. Differ. Equ. 15, No. 9-10, 893-924 (2010).

Summary: 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
\[ \begin{aligned} -\Delta_pu+c(x)|u|^{p-2}u= f &\quad\text{in }\Omega,\\ d{\mathbf N}_p(u)+\beta(x,u)d\mu= gd\mu &\quad\text{on }\partial\Omega, \end{aligned} \]
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 \({\mathbf N}_p(u)\) is a generalization of the normal derivative for bad domains. When \(\Omega\) and \(u\) are smooth, then \(d{\mathbf N}_p(u)= |\nabla u|^{p-2}(\partial u/\partial\nu)d\sigma\) where \(\sigma\) is the surface measure and v the outer normal to \(\partial\Omega\). A priori estimates for solutions are also obtained.

MSC:

35J62	Quasilinear elliptic equations
35B45	A priori estimates in context of PDEs
35D30	Weak solutions to PDEs
35J25	Boundary value problems for second-order elliptic equations

Keywords:

quasi-linear elliptic equations; Robin boundary conditions; Borel measure; a priori estimates