Mixed boundary value problems for nonlinear elliptic equations with \(p\)-structure in nonsmooth domains
The authors prove regularity results for weak solutions of the following nonlinear mixed boundary value problems: \[ \begin{aligned} -\sum^n_{i=1}\partial_iF_i(x,\nabla u) & =f(x)+\sum^n_{i=1}\partial_if_i(x)\text{ in }\Omega;\\ x(u) & = 0 \text{ on }\Gamma_D;\\ -\sum^n_{i=1}F_i(x,\nabla u)\nu_i(x) & =\sum^n_{i=1}f_i(x)\nu_i(x)\text{ on }\Gamma_N.\end{aligned} \tag{*} \] Here \(\Omega\subset \mathbb{R}^n\) is a polyhedral domain, \(\nu\) the outward unit normal of \(\partial\Omega\) and \(\partial\Omega = \Gamma_D\cup \Gamma_N\) with \(\Gamma_D\neq\emptyset\). Moreover, \(\partial\Omega\) is assumed to be a union of ``nonoverlapping'' polyhedral subsets of hyperplanes, which belong to either \(\Gamma_D\) or \(\Gamma_N\). The inner angle between two such adjacent pieces is strictly between 0 and \(2\pi\) in general and between 0 and \(\pi\), if Neumann and Dirichlet pieces border on each other. As for the elliptic principal part one may think of the \(p\)-Laplace operator, i.e. \(F_i(x,\nabla u) = |\nabla u|^{p-2}\partial_i u\), where it is assumed that \(1 < p\geq n-1\). Under further suitable assumptions on \(f_i\), and \(f\) it is proved that there exists a positive number \(\gamma\) independent of \(n\) such that any weak solution \(u\in W^{1,p}(\Omega)\) of (*) has the additional regularity: \[ \nabla u \in L^q(\Omega)\text{ for }g=p+1+\gamma. \] Regularity results are also shown in Morrey spaces.
scientific article; zbMATH DE number 1832843