Weakly elliptic systems of variational inequalities: A \(2 \times 2\) model problem with obstacles in both components
The authors solve the problem with the obstacle \(\psi= (\psi_1, \psi_2): \Omega\to \mathbb{R}^2\) such that \(\psi_i< 0\) near \(\partial \Omega\) for \(i= 1,2\): Find \(U\in K\) such that \(\langle A\Delta U- BU, V- U\rangle\geq 0\) for all \(V\in K\), \(K= \{W\in H^1_0(\Omega)\times H^1_0(\Omega)\mid W_i\geq \psi_i\), \(i= 1,2\}\), \(A\) and \(B\) are \(2\times 2\) matrices. Various assumptions and types of positivity of the matrix \(-A\) are considered. Using the penalization method, existence theorems for systems which are not strongly elliptic are obtained. An example of non-uniqueness in the case of \(\Omega= B(0, 1)\) the unit ball in \(\mathbb{R}^n\), \(A= (\begin{smallmatrix} 0\\ -1\end{smallmatrix} \begin{smallmatrix} -1\\ 0\end{smallmatrix})\) and \(B= 0\) is given.
scientific article; zbMATH DE number 878429
scientific article