On classical solvability of a non-local boundary value problem for an elliptic equation of the second order. I (Q1902847)

Let \(D\) be a bounded domain in \(\mathbb{R}^n\) with boundary \(S\) of class \(C^2\). Assume that there exists a single valued mapping of \(S\) onto the set \(\sigma S\subset\mathbb{R}^n\) and write \(\omega=\sigma S\cap S\), \(\widehat\omega=\{x\in S:\sigma x\in\omega\}\). Boundary value problems of the type indicated in the title which relate the traces of the required solution and its derivatives on \(S\) and on a set \(S'\subset\overline D=D\cup S\) with \(S'\cap S=\emptyset\) have been well studied. The case \(S'\cap S\neq\emptyset\) is considered in this paper. Consequently, it is assumed throughout the paper that \(\omega\neq\emptyset\), and \(\omega\neq S\). Under these conditions, a model boundary value problem, together with its homogeneous counterpart, is defined. Technical conditions, depending on \(\sigma\), \(\omega\) and the differential expression involved, are then introduced which enable existence and uniqueness results to be obtained in a number of normed linear spaces. Estimates of the solution are also presented.