The Dirichlet problem for quasilinear elliptic equations in domains with smooth closed edges (Q2773585)

The authors consider the Dirichlet problem for the second-order elliptic equation NEWLINE\[NEWLINE -\sum_{i,j=1}^n a_{ij}(x,u,Du)D_iD_j u + a(x,u,Du)=0,\;\;x\in \Omega. NEWLINE\]NEWLINE The boundary of a bounded domain \(\Omega\) may contain a conical point or an edge of arbitrary dimension. The solvability questions are studied in the weighted Sobolev spaces. The weight depends on a power of the distance from a given point to this edge or conical point. The solvability theorem relies on the corresponding linear theory and a priori bounds which make it possible to apply the method of continuation in a parameter. In particular, Hölder estimates for a solution and the gradient of a solution are established.