Interface problems for quasilinear elliptic equations
Let \(\Omega\subset{\mathbb R}^2\) be a polygonal domain whose closure is the union of the closures of finitely many polygonal subdomains \(\Omega^{(k)}\). The author studies the uniformly elliptic Dirichlet problem \[ -D_i[A^{ij}(x,u)D_j]=-D_iF^I\quad \text{on }\Omega, \qquad u|\partial\Omega=0 \] assuming that \(A^{ij}|[\text{cl}(\Omega^{(k)})\times{\mathbb R}]\in C^1(\text{cl}(\Omega^{(k)})\times{\mathbb R})\) and \(F^i\in H^1(\Omega)\cap C^{0,\sigma}(\Omega)\). Roughly speaking, he establishes \(H^2\cap C^{1,\sigma}\)-interior regularity in \(\Omega^{(k)}\) for bounded weak solutions and obtains then a representation of a solution near a singular point in terms of particular solutions.
scientific article; zbMATH DE number 1346394
scientific article