Asymptotic of a solution of the Neumann problem at a point of tangency of smooth components of the boundary of the domain

S. A. Nazarov

Izvestiya. Mathematics , Tome 44 (1995) no. 1, pp. 91-118

Résumé

The asymptotics of the solution of the Neumann problem is studied for a second-order elliptic equation near a point of tangency of two surfaces forming the boundary of a domain in $\mathbf R^n$, $n\geqslant 3$. In accordance with the procedure of investigating problems in thin domains, the resulting equation is found on the hyperplane $\mathbf R^{n-1}$, the power solutions of which occur in the asymptotics. The justification of the expansion first found formally is based on a priori estimates of solutions in spaces with weighted norms, reduction of the problem to the resulting equation by means of integration, and application of a familiar theorem regarding the asymptotics of the latter.