Investigation of a boundary value problem in a plane infinite wedge for the Laplacean with the boundary condition of a special type

V. A. Solonnikov ; E. V. Frolova

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 21, Tome 182 (1990), pp. 149-167

Résumé

We construct explicitly and estimate in weighted S. L. Sobolev spaces the solution of the equation $\Delta u=f$ in a plane infinite wedge satisfying the Neumann condition on one side of the wedge and the condition $\frac{\partial u}{\partial n}+h\frac{\partial u}{\partial r}+\sigma u=\psi$ on another side ($\frac\partial{\partial r}$ is the tangential derivative, $\sigma\in\mathbb{C}$, $\mathrm{Re}\,\sigma\geqslant0$). Our estimates are exact with respect to the differential order and uniform with respect to $\sigma$. The construction of the solution reduces after the Mellin transform to the investigation of a finite difference equation on the complex plane.