boundary values in Lebesgue spaces with \(p >1\)
The Dirichlet problem for a second-order elliptic equation with an \(L_{p}\) boundary function
The main result of this paper is a unique solvability theorem for a Dirichlet problem where the boundary value lies in the Lebesgue spaces with \(p >1\), in a bounded domain in \(\mathbb R^n\). The weighted Sobolev spaces are involved, the weight being the distance from the boundary. The inward pointing normal on the boundary is assumed to be Dini continuous; the same property is also assumed for the equation coefficients, on the domain boundary. A very interesting example is given in the last part, concerning the possibility of a weaker condition for the uniqueness of the solution.
scientific article; zbMATH DE number 6039339