Elliptic boundary value problem with superposition operator in the boundary condition. I (Q1589206)

The paper deals with the classical solvability of the problem \[ \left\{\begin{aligned} &\sum_{i,j=1}^n a_{ij}(x)u_{x_ix_j} +\sum_{i=1}^n b_i(x)u_{x_i} +c(x) u=f(x)\quad x\in D,\\ &u(x)-u(\sigma x)=\psi(x)\quad x\in \partial D,\end{aligned}\right. \] where \(D\) is a bounded and smooth domain in \({\mathbb{R}}^n,\) the second-order differential operator is supposed to be strictly elliptic with Hölder continuous coefficients, \(c(x)\leq 0,\) and \(\sigma: {\mathbb{R}}^n\to {\mathbb{R}}^n\) is a one-to-one mapping. The unique classical solvability is linked to the solvability of the equation \[ u(x)-u(\sigma x)=\psi(x)\qquad x\in \sigma(\partial D)\cap \partial D \] assuming that \(\sigma(\partial D)\cap D\subset \overline{D}\) and \(\sigma(\partial D)\cap \partial D\not=\emptyset.\)