Boundedness and continuity of solutions to linear elliptic boundary value problems in two dimensions
The paper deals with mixed boundary value problems for linear second order elliptic differential equations in a bounded two-dimensional Lipschitzian domain \(\Omega\). It is shown that the solutions to such problems are bounded and continuous provided that the right hand side \(g\) of the equation is such that \(g\log(1+ | g|)\in L^ 1(\Omega)\) and that the right hand side of the natural boundary condition has a similar property. The proof is based on results on the solvability of boundary value problems in the context of Sobolev spaces and on the introduction of some new interpolation spaces.
linear second order elliptic differential equations
bounded two-dimensional Lipschitzian domain
scientific article; zbMATH DE number 562484