Poisson Problem for a Functional–Differential Equation. Positivity of a Quadratic Functional. Jacobi Condition

Abstract

For the Poisson problem

$$\begin{aligned} \mathcal {L}u := -\varDelta u + p(x)u - \int \limits _\varOmega u(s)\,r(x,\mathrm{d}s) = \rho f, \quad u\big |_{\varGamma (\varOmega )} =0 \end{aligned}$$

equivalence of positivity of the quadratic functional

$$\begin{aligned} \int \limits _\varOmega u'_xu'_x \,\mathrm{d}x + \int \limits _\varOmega p(x) u(x)^2\,\mathrm{d}x - \int \limits _{\varOmega \times \varOmega }u(x) u(s)\,\xi (\mathrm{d}x\times \mathrm{d}s), \end{aligned}$$

($\mathrm{d}x := \mathrm{d}x_1\cdots \mathrm{d}x_n$), the corresponding Jacobi condition, and positivity of the Green function are showed.