Boundedness and regularity of solutions of degenerate elliptic partial differential equations
The paper under review deals with suitably defined weak solutions to the equation \[ \sum_{i,j}\frac{\partial}{\partial x_i}\left(B_{ij}(x,u,\nabla u)\frac{\partial}{\partial x_j}u\right)+g(x,u,\nabla u)=0, \] where \(x\) belongs to an open set \(\Omega\) in an \(n\)-dimensional metric space \(X\) and the \(n\times n\) matrix-valued function \(B=\{B_{ij}\}\) is nonnegatively definite. By applying the Moser iterations, the author proves local boundedness of the solutions when a weighted Sobolev embedding holds on \(X\). Moreover, a Harnack type inequality for the nonnegative solutions is obtained assuming a weighted version of the Poincaré for their logarithm.
scientific article; zbMATH DE number 6756298
scientific article