The Hardy's inequality and positive invertability of elliptic operators (Q2735884)

The paper is devoted to positive invertability of elliptic operators of the form \((Au)(x)= -\Delta u(x)- a(x)u(x)\) on the domain \(D(A)= \overset\circ W_p^1(\Omega)\cap w_p^2 (\Omega)\). This result is based on exact value of generalized Hardy's constant. Let us define NEWLINE\[NEWLINE\begin{aligned} l_{p,\Omega}(u) &= \Biggl[ \int_\Omega|\nabla u|^p\,dx \Biggr]^{\frac{1}{p}} \cdot \Biggl[ \int_\Omega| u|^p (\text{dist} (x,\partial\Omega))^{-p}\,dx \biggr]^{-\frac{1}{p}},\\ H_{p,\Omega} &= \inf\{l_{p,\Omega}(u),\;u\in \overset\circ W_p^1(\Omega),\;u\not\equiv 0\}. \end{aligned}NEWLINE\]NEWLINE It is proved that if \(\max_\Omega \{[a(x) ]^{1/2}\cdot \text{dis} (x,\partial\Omega) \}< H_{2,\Omega}\), then for any \(p\in (1,+\infty)\) there exists the bounded positive inverse \(A^{-1}\) in \(L_p\) and the estimate \(\|\exp (\{-tA \}\|_{L_{p(\Omega)}\to L_{p(\Omega)}}\leq M_p e^{-\varepsilon \lambda,t}\), \(t\geq 0\) is true for some \(1\leq M_p< +\infty\) and \(\varepsilon= 1-H_{2,\Omega}^{-2}\cdot \max_{x\in\Omega} \{[a(x) ]_+\cdot \text{dist}^2 (x,\partial \Omega)\}\).