Study of Poincaré-Hardy type inequalities and eigenvalue problems for second-order elliptic PDEs

Abstract

The major text of this thesis is studying Poincaré-Hardy and Hardy-Rellich type inequalities on one of the most discussed Cartan-Hadamard manifold namely hyperbolic space and studying eigenvalue problems for second-order elliptic PDEs. The thesis is divided into two parts. In the first part we have centralized our attention on the following three problems: • On some strong Poincaré inequalities on Riemannian models and their improvements. • On higher order Poincaré inequalities with radial derivatives and Hardy improvements on the hyperbolic space. • Hardy-Rellich and second order Poincaré identities on the hyperbolic space via Bessel pairs. In the second part we have focused our essence on the following two problems: • Generalized principal eigenvalues of convex nonlinear elliptic operators in RN. • On ergodic control problem for viscous Hamilton-Jacobi equations for weakly coupled elliptic systems.