Normal solvability of the Dirichlet problem for the vibrating string equation in domains with nonsmooth boundaries
The article is devoted to the Dirichlet problems in a bounded convex domain \(\Omega \subset \mathbb{R}^ 2\) with piecewise smooth boundary for the equations \(\overline {\mathcal L}_ \lambda u = f\) and \(\overline {\mathcal L}_ \lambda^* v = g\) \((f,g \in L_ 2)\), where \({\mathcal L}_ \lambda u = (1 + \lambda) u_{xx} - (1 - \lambda) u_{yy}\) \((\lambda \in (- 1,1))\). The main goal is to study the winding number of some auxiliary homeomorphism \(F_ \lambda\) that is defined with \(\Omega\); in particular necessary and sufficient conditions are given under which the winding number is rational for all \(\lambda\) or for all, except a finite number of \(\lambda\)'s. As a corollary some results on normal solvability for the problems above are presented.
scientific article; zbMATH DE number 130810