Maximum and antimaximum principles for some nonlocal diffusion operators (Q1044487)

Let \(J\) be a continuous function in \(\mathbb{R}^N \) under restrictions: \(J(x)=0\) if \(\|x\|\geq 1\), \(J(x) \geq 0\), \(J(x)=J(-x)\), \(J(x) \not\equiv 0\). Let \(\Omega \) be a bounded domain in \(\mathbb{R}^N \). It is known that there is the unique value \(\lambda = \lambda_1(\Omega)\) such that for some function \(u(x)\geq 0\), \(u(x) \not\equiv 0\), \(u\in L_2(\mathbb{R}^N )\) the following condition is fulfilled: \( ( J\ast u-u )(x)= \lambda u(x)\) if \(x\in \Omega\), \(u(x)=0\) if \(x\not\in \Omega\). The function \(u\) is defined by this condition uniquely. This function is denoted by \(\varphi (x)\) in what follows. The authors study the problem \[ \begin{cases}\int_{\mathbb{R}^N}J(x-y)u(y)dy -u(x)+\lambda u(x) +h(x) =0,& x\in \Omega \\ u(x)=0, & x\not\in \Omega \end{cases}\tag{1} \] They prove such results: 1. If \(h\in L_2(\Omega)\), \(h\geq 0\), \(h \not\equiv 0\), \(\lambda < \lambda_1(\Omega)\), \(u\in L_2(\mathbb{R}^N )\) is the solution to (1) then \(u>0\) in \(\overline{\Omega}\) (\(ess\inf \{ u(x): x\in \Omega \}>0\)). 2. If \(h\in L_{\infty}(\Omega)\), \(\int_{\Omega}h(x)\varphi(x)dx >0\) then there exists \(\varepsilon(h)>0\) such that for \(\lambda \in (\lambda_1(\Omega), \lambda_1(\Omega) +\varepsilon)\) the unique solution \(u\) to (1) verifies \(u<0\) in \(\overline{\Omega}\). There are many other results in the work. Applications are discussed as well. There is detail survey of related results.