Ph.D. Thesis
"Study of the Laplacian in a Class of Doubly Connected Domains on the Riemann Sphere $S^2$"
Abstract.
Fix an open ball $B_1$ of radius $r_1$ in $S^n$ ($H^n$). Let $B_0$ be any open ball of radius $r_0$ such that $\overline{B_0} \subset B_1$. For $S^n$ we consider $r_1< \pi$. Let $u$ be a solution of the problem $\Delta u =1$ in $\Omega := B_1 \setminus \overline{B_0}$ vanishing on the boundary. It is shown that the energy functional is minimal if and only if the balls are concentric. It is also shown that first Dirichlet eigenvalue of the Laplacian on $\Omega$ is maximal if and only if the balls are concentric.