On a~class of~elliptic boundary-value problems with parameter and discontinuous non-linearity

V. N. Pavlenko ; D. K. Potapov

Izvestiya. Mathematics , Tome 84 (2020) no. 3, pp. 592-607

Résumé

We study an elliptic boundary-value problem in a bounded domain with inhomogeneous Dirichlet condition, discontinuous non-linearity and a positive parameter occurring as a factor in the non-linearity. The non-linearity is in the right-hand side of the equation. It is non-positive (resp. equal to zero) for negative (resp, non-negative) values of the phase variable. Let $\widetilde{u}(x)$ be a solution of the boundary-value problem with zero right-hand side (the boundary function is assumed to be positive). Putting $v(x)=u(x)-\widetilde{u}(x)$, we reduce the original problem to a problem with homogeneous boundary condition. The spectrum of the transformed problem consists of the values of the parameter for which this problem has a non-zero solution (the function $v(x)=0$ is a solution for all values of the parameter). Under certain additional restrictions we construct an iterative process converging to a minimal semiregular solution of the transformed problem for an appropriately chosen starting point. We prove that any non-empty spectrum of the boundary-value problem is a ray $[\lambda^*,+\infty)$, where $\lambda^*>0$. As an application, we consider the Gol'dshtik mathematical model for separated flows of an incompressible fluid. We show that it satisfies the hypotheses of our theorem and has a non-empty spectrum.

Détail

Keywords: elliptic boundary-value problem, problem with parameter, discontinuous non-linearity, iterative process, minimal solution, semiregular solution, spectrum, Gol'dshtik model.