Abstract. We study Hardy-type inequalities associated to the quadratic form of the shifted Laplacian −∆ H N − (N − 1) 2 /4 on the hyperbolic space H N , (N − 1) 2 /4 being, as it is well-known, the bottom of the L 2 -spectrum of −∆ H N . We find the optimal constant in a resulting Poincaré-Hardy inequality, which includes a further remainder term which makes it sharp also locally: the resulting operator is in fact critical in the sense of [17]. A related improved Hardy inequality on more general manifolds, under suitable curvature assumption and allowing for the curvature to be possibly unbounded below, is also shown. It involves an explicit, curvature dependent and typically unbounded potential, and is again optimal in a suitable sense. Furthermore, with a different approach, we prove Rellich-type inequalities associated with the shifted Laplacian, which are again sharp in suitable senses.

We consider a partially hinged rectangular plate and its normal modes. The dynamical properties of the plate are influenced by the spectrum of the associated eigenvalue problem. In order to improve the stability of the plate, it seems reasonable to place a certain amount of stiffening material in appropriate regions. If we look at the partial differential equation appearing in the model, this corresponds to insert a suitable weight coefficient inside the equation. A possible way to locate such regions is to study the eigenvalue problem associated to the aforementioned weighted equation. In the present paper we focus our attention essentially on the first eigenvalue and on its minimization in terms of the weight. We prove the existence of minimizing weights inside special classes and we try to describe them together with the corresponding eigenfunctions.

We investigate the possibility of improving the p-Poincaré inequality ∇ H N u p p ≥ Λp u p p on the hyperbolic space, where p > 1 and Λp := [(N −1)/p] p is the best constant for which such inequality holds. We prove several different, and independent, improved inequalities, one of which is a Poincaré-Hardy inequality, namely an improvement of the best p-Poincaré inequality in terms of the Hardy weight r −p , r being geodesic distance from a given pole. Certain Hardy-Maz'ya-type inequalities in the Euclidean half-space are also obtained.

We prove optimal improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of [19], namely the associated inequality cannot be further improved. Such inequalities arise from more general, optimal ones valid for the operator P λ := −∆ H N − λ where 0 ≤ λ ≤ λ1(H N ) and λ1(H N ) is the bottom of the L 2 spectrum of −∆ H N , a problem that had been studied in [5] only for the operator P λ 1 (H N ) . A different, critical and new inequality on H N , locally of Hardy type, is also shown. Such results have in fact greater generality since there are shown on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator P λ .In case of Cartan-Hadamard manifold M of dimension N (namely, a manifold which is complete, simply-connected, and has everywhere non-positive sectional curvature), the geodesic distance function d(x, x 0 ), where x 0 ∈ M , satisfies all the assumptions of the weight ̺ and the above inequality holds with best constant N −2 2 2 , see [31]. In particular, consid-Date: November 23, 2017. 2010 Mathematics Subject Classification. 46E35, 26D10, 31C12.

In this article we will study the existence and nonexistence of sign changing solutions for the Brezis-Nirenberg type problem in the Hyperbolic space. We will also establish sharp asymptotic estimates for the solutions and the compactness properties of solutions.