Elliptic boundary value problem with superposition operator in the boundary value condition. II.
The article continues previous work of the author [Part I, Russ. Math. 44, No. 4, 34--38 (2000); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2000, No. 4, 36--40 (2000; Zbl 0965.35032)] where the Bitsadze--Samarskii problem \[ {\mathcal L}u\equiv \sum_{i,j=1}^n a_{ij}(x)u_{x_ix_j} + \sum_{i=1}^n b_{i}(x)u_{x_i}+c(x)u=f(x)\qquad x\in D\subset {\mathbb R}^n, \] \[ {\mathcal B}u\equiv u(x)-u(\sigma x)=\psi(x)\qquad x\in S=\partial D \] was considered under the assumption that the set \(\sigma S \cap S\equiv \omega\) is nonempty and \(\sigma\omega\subset\omega.\) Here the problem is studied without these constraints on \(\sigma.\) Precisely, the solvability theory depends essentially on the iterates of the mapping \(\sigma,\) that is, whether \(\sigma\) generates the ``absorbing set'' \( \Omega\equiv \left\{x\in S: \sigma^i x\in S,\;i=1,2,\ldots\right\}\) on \(S\).
scientific article; zbMATH DE number 1956410