Regularity of the solution of a variational inequality for two second order elliptic operators
The authors consider a Lipschitz bounded open subset \(\Omega\) of \({\mathbb{R}}^ n\) and two uniformly elliptic operators \(A_ k\) \((k=1,2)\) given by (with the usual summation convention) \[ A_ k=-D_ j(a^ k_{ij}(x)D_ i)+b^ k_ i(x)D_ i+c^ k(x) \] where (for \(n\geq 3)\) \(a^ k_{ij}\in L^{\infty}(\Omega)\), \(b^ k_ i\in L^ n(\Omega)\), \(c^ k\in L^{n/2}(\Omega)\). Under the assumption that the bilinear forms \[ <A_ ku,v>=\int_{\Omega}[a^ k_{ij}(x)D_ iuD_ jv+b^ k_ i(x)D_ iuv+c^ k(x)uv]dx \] are coercive in \(H^ 1_ 0(\Omega)\), the variational inequality \[ (*)\quad <A_ 1u_ 1,v_ 1- u_ 1>+<A_ 2u_ 2,v_ 2-u_ 2>\geq <f_ 1,v_ 1-u_ 1>+<f_ 2,v_ 2-u_ 2>,\quad (u_ 1,u_ 2)\in K,\quad (v_ 1,v_ 2)\in K \] is considered, where \(K=\{(v_ 1,v_ 2)\in (H^ 1_ 0(\Omega))^ 2: v_ 1+\psi \geq v_ 2\) a.e. on \(\Omega\) \(\}\), \(f_ i\in H^{- 1}(\Omega)\), and \(\psi \in H^ 1(\Omega)\) with \(\psi\geq 0\) on \(\partial \Omega\). The paper is devoted to the study of the regularity properties of the solution of (*).
scientific article; zbMATH DE number 3933870