Laplace operator with \(\delta\)-like potentials
Let \(\Omega \) be a simply connected domain with smooth boundary in \(\mathbb R^2\), and \(M=[x_0,y_0]\) be a fixed inner point of the domain \(\Omega \). Denote \(\Omega_0 =\Omega \setminus \{ M\} \). Denote by \(G(x,y,x_0,y_0)\) the Green function of the Dirichlet problem for the Laplace equation corresponding to \(\Omega \) and \([x_0,y_0]\). The Laplace operator is studied on the space \(\{ h(x,y)=h_1(x,y)+\alpha G(x,y,x_0,y_0);\alpha \in\mathbb R, h_1\in W^{2,2}(\Omega ), h_1 =0\) on \(\partial \Omega \} \). An analog of the Green formula is obtained.
scientific article; zbMATH DE number 6366137
scientific article