A pointwise comparison for solutions of linear elliptic equations
Comparison theorems are proved for solutions of two Dirichlet problems for second-order linear elliptic equations (1) \(Lu=f\) in a bounded domain \(G\subset {\mathbb{R}}^ n\), \(n\geq 2\). A weak solution \(u\in W_ 0^{1,2}(G)\) of (1) is compared with the solution v of a problem with radial symmetry. In particular, a pointwise comparison \(u^{\#}(x)\leq v(x)\) is obtained a.e. in a ball, where \(u^{\#}\) denotes the symmetric decreasing rearrangement of u.
scientific article; zbMATH DE number 4013135