A minimization problem with free boundary and its application to inverse scattering problems (Q6593762)

Under consideration is the problem of minimizing the functional \N\[\NJ(u)=\int_G (|\nabla u|^2-\lambda |u|^2 -2f(x)u -g^2(x)\chi_{u>0})\,dx\N\] \Nwhere the function \(u\in W_2^1(G)\) satisfies the homogeneous Dirichlet boundary condition and \(G\) is a \(C^1\)-domain in \(\mathbb{R}^n\). It is proven that if \(\lambda<\lambda^*\) (\(\lambda^*\) is the first eigenvalue of the operator \(-\Delta\) with the Dirichlet boundary conditions) then the minimizer of this functional exists. Some applications to inverse scattering problems are given.