Smooth continuation of solutions and eigenvalues for variational inequalities based on the implicit function theorem.
The authors obtain interesting new results concerning continuation of solutions of variational inequalities depending on an eigenvalue parameter. The proof of the main result is based on an implicit function theorem applied to an equation which is equivalent to the variational inequality under consideration. As an application, the authors consider a class of elliptic equations of the type \(\triangle u(x)+\lambda(x)u(x)=\mu u(x)\) in \(\Omega\) with boundary conditions \(u=0\) on \(\Gamma_D,\;\partial u/\partial n=0\) on \(\partial\Omega\setminus (\Gamma_D\cup\Gamma_U),\;\int_{\Gamma_U} u\;d\Gamma\geq0,\;\partial u/\partial n\geq0\) on \(\Gamma_U,\;\partial u/\partial n\) is constant on \(\Gamma_U\) and \(\int_{\Gamma_U} u\;d\Gamma\cdot \partial u/\partial n=0\) on \(\Gamma_U\).
scientific article; zbMATH DE number 1866621
scientific article