Dirichlet problems for singular elliptic equations

Abstract:

Boundary value problems are formulated for the equation \[ ( \ast )\quad L[u] = \sum \limits _{i,j = 1}^n {{a_{ij}}\frac {{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}}} + \sum \limits _{i = 1}^{n - 1} {{b_i}\frac {{\partial u}}{{\partial {x_i}}}} + \frac {h}{{{x_n}}}\frac {{\partial u}}{{\partial {x_n}}} + cu = f\] in a bounded domain $G$ in ${E_n}$ with boundary $\partial G = {S_1} \cup {S_2}$ where ${S_1}$ is in ${x_n} = 0$ and ${S_2}$ is in ${x_n} > 0$. A uniqueness theorem is established for $( \ast )$ when boundary data is only given on ${S_2}$ for \[ h({x_1}, \cdots ,{x_{n - 1}},0) \geqq 1;\]; whereas an existence and uniqueness theorem for the Dirichlet problem is proved for $h({x_1},{x_2}, \cdots ,{x_{n - 1}},0) < 1$.