On some noncoercitive variational inequalities (Q2784867)

The authors study existence and regularity of solutions of two variational inequalities. The first one has the form: NEWLINE\[NEWLINE\sum_{k=1}^{2}\sum_{i,j=1}^{n}\int_{\Omega_k}a_{ij}^k(u_k)'_{x_i} (v_k-u_k)'_{x_j}+\int_{\Omega_1}au_1(v_1-u_1) dx\geq \sum_{k=1}^{k}\langle f_k,v_k-u_k\rangle_k \quad \forall(v_1,v_2)\in K.NEWLINE\]NEWLINE Here \(\Omega_1,\Omega_2\) are such open sets of \(\mathbb R^n\) that either \(\Omega_1=\Omega_2\) or \(\overline{\Omega_2}\subset\Omega_1\), \(V_k\) denotes either \(H^1(\Omega_k)\) or \(H^1_0(\Omega_k)\), the matrices \(\{a_{i,j}^k(x)\}_{i,j=1}^n\) determine coefficients of uniformly elliptic operators, \(a(x)\) is an essentially bounded and uniformly positive function on \(\Omega_1\), \(K\) is the set of such \((v_1,v_2)\in V_1\times V_2\) that \(v_1\geq v_2\) a.e. on \(\Omega_2\), \(f_1,f_2\) are given elements of \(V_1^*\) and \(V_2^*\), respectively. The authors establish necessary conditions for the existence of a solution \((u_1,u_2)\in K\) and analyze the case when the solution is unique and \(H^2\)-regular.NEWLINENEWLINENEWLINEThe second inequality is less traditional and, in the authors' opinion, is studied at the first time.