Elliptic equations, rearrangements, and functions of bounded lower oscillation
Let \(\Omega\) be a bounded domain in \({\mathbb{R}}^ n\) and H(x), \(w_ 1(x)\), \(w_ 2(x)\) nonnegative functions on \(\Omega\) with \(H\in L^{\infty}(\Omega)\), \(0<\alpha \leq H(x)\), and \(w_ 1,w_ 2\in L^ 2(\Omega)\). Let \(H^{\#}\), \(w_ 1^{\#}\) and \(w_ 2^{\#}\) denote the radial decreasing rearrangements of H, \(w_ 1\) and \(w_ 2\) on a ball \(\Omega^{\#}\) in \({\mathbb{R}}^ n\) with mes \(\Omega^{\#}=mes \Omega\). Let \(u\in H^ 1_ 0(\Omega)\) and \(v\in H^ 1_ 0(\Omega^{\#})\) be the weak solutions of the equations \[ (1)\quad - div(H(x)\nabla u)=w_ 1\quad in\quad \Omega;\quad u=0\quad on\quad \delta \Omega, \] \[ (2)\quad -div(H^{\#}(x)\nabla v)=w_ 1^{\#}\quad in\quad \Omega^{\#};\quad v=0\quad on\quad \delta \Omega^{\#}. \] The authors construct a class of functions W(\(\Omega)\) with the property: Theorem 1. If \(w_ 1,w_ 2\in W(\Omega)\) and u and v satisfy equations (1) and (2) respectively, then the following estimate holds: \[ \int_{\Omega}uw_ 2 dx\leq \int_{\Omega^{\#}}vw_ 2^{\#} dx. \] The authors also show that the class W(\(\Omega)\) is the best possible.
scientific article; zbMATH DE number 4120651
scientific article