The modified Dirichlet problem for the elliptic system of equations that degenerates on the $n$-dimensional sphere and at the origin of coordinates

G. A. Isaeva

Fundamentalʹnaâ i prikladnaâ matematika, Tome 2 (1996) no. 4, pp. 1205-1212.

Résumé

The belonging of a system of partial differential equations with variable coefficients to one or another homotopic type depends on the domain point at which this system is considered. The degeneration manifolds split the original region into parts. The study of the influence of such degeneration on the solvability character of the boundary value problems is important. We consider the system of $n$ partial second order differential equations $$ -(x_1^2+x_2^2+\ldots+x_n^2)\Delta u_j+\lambda\frac{\partial}{\partial x_j}\sum_{i=1}^{n}\frac{\partial u_i}{\partial x_i}=0,\quad j=1,\ldots,n, $$ with real parameter $\lambda>0$. This system is elliptic everywhere, except the origin of coordinates and the $n$-dimensional sphere with radius $\sqrt{\lambda}$, on which the parabolic degeneration occurs. We prove that the modified Dirichlet problem for this system considered within a ball that either contains the degeneration sphere or is situated inside it, is solvable, and the solution is unique in the class of bounded functions.