On positivity of the Green function for Poisson problem for a linear functional differential equation

S. M. Labovski

Vestnik rossijskih universitetov. Matematika, Tome 22 (2017) no. 6, pp. 1229-1234

Résumé

For the Poisson problem \begin{equation*} -\Delta u + p(x)u - \int\limits_\Omega u(s)\,r(x,ds) = \rho f, \quad u\big|_{\Gamma(\Omega)} =0 \end{equation*} equivalence of positivity of the Green function and other classical properties is showed. Here $\Omega$ is an open set in $\mathbb{R}^n$, and $\Gamma(\Omega)$ is the boundary of the $\Omega$. For almost all $x\in\Omega$, $r(x,\cdot)$ is a measure satisfying certain symmetry condition. In particular this equation involves integral differential equation and the equation $$ -\Delta u + p(x)u(x) - \sum_{i=1}^{m}p_i(x)u(h_i(x)) = \rho f, $$ where $h_i\colon \Omega\to\Omega$ is a measurable mapping.

Détail

Keywords: Green function, Vallee-Poussin theorem, Spectrum of selfadjoint operator.
Mots-clés : Poisson problem