Dirichlet problem for degenerate elliptic equations

Abstract:

Let ${L_0}$ be a degenerate second order elliptic operator with no zeroth order term in an m-dimensional domain G, and let $L = {L_0} + c$. One divides the boundary of G into disjoint sets ${\Sigma _1},{\Sigma _2},{\Sigma _3};{\Sigma _3}$ is the noncharacteristic part, and on ${\Sigma _2}$ the “drift” is outward. When c is negative, the following Dirichlet problem has been considered in the literature: $Lu = 0$ in G, u is prescribed on ${\Sigma _2} \cup {\Sigma _3}$. In the present work it is assume that $c \leq 0$. Assuming additional boundary conditions on a certain finite number of points of ${\Sigma _1}$, a unique solution of the Dirichlet problem is established.