Uniqueness results for optimzation problems with prescribed rearrangement
The paper is concerned with comparison results for the solutions of elliptic Dirichlet problems as well as Hamilton-Jacobi equations in a domain \(\Omega\) in \(\mathbb{R}^n\) and the solutions of the corresponding symmetric problems. In particular, the authors investigate the case when equality occurs for \(L^\infty\) or \(L^1\) norms in such comparison. A typical result is the following. Let \(f\in L^p(\Omega)\), \(\nu>0\), \(\nu^{-1}\in L^t\), \(1/t+1/p<2/n\), and let \(u\in W_0^{1,p}(\nu,\Omega)\) satisfy the equation \(-(a_{ij}u_i)_j=f\) in \(\Omega\), where \(a_{ij}(x)\xi_i\xi_j\geq \nu|\xi|^2\) for all \(\xi\in\mathbb{R}^n\). Let \(\nu_\#\) be the increasing spherically symmetric rearrangement of \(\nu\), let \(f^\#\) be the decreasing spherically symmetric rearrangement of \(f\), let \(\Omega^\#\) be a ball such that \(|\Omega^\#|= |\Omega|\), and let \(v\in W^{1,2}(\nu_\#,\Omega^\#)\) satisfy the equation \(-(\nu_\# v_i)_i=f^\#\) in \(\Omega^\#\). If \(|u|_\infty=|v|_\infty\) then it must be \(\Omega=\Omega^\#\), \(u=\mp u^\#\), \(f=\mp f^\#\), \(\nu=\nu_\#\) and \(a_{ij}x_j=\nu x_i\) modulo translations. A similar result is given when \(n=2\), assuming \(|u|_1=|v|_1\). Furthermore, the torsion problem, the capacity problem and the Hamilton-Jacobi equation are discussed. In all cases the authors prove that equality in the comparison of certain norms is achieved only in spherically symmetric situations.
scientific article; zbMATH DE number 903386
scientific article