Title: Thespace of Hardy-weights for quasilinear equations: Maz’ya-type characterization and sufficient conditions for existence of minimizers.

Abstract: Let $p \in (1,\infty)$ and  $\Omega\subset \mathbb{R}^N$ be a domain. Let  $A: =(a_{ij}) \in L^{\infty}_{loc}(\Omega;\mathbb{R}^{N\times N})$ be  a symmetric and locally uniformly positive definite matrix. Set $|\xi|_A^2:= \sum_{i,j=1}^N a_{ij}(x) \xi_i \xi_j$, $\xi \in \mathbb{R}^N$, and let $V$ be a real valued potential in a certain local Morrey space. We assume that the energy functional 
$$Q_{p,A,V}(\phi):=\int_{\Omega} (|
abla \phi|_A^p + V|\phi|^p) dx $$
is nonnegative on $W^{1,p}(\Omega)\cap C_c(\Omega)$.
   
We introduce a generalized notion of $Q_{p,A,V}$-capacity and characterize  the space of all Hardy-weights for the functional $Q_{p,A,V}$, extending Maz'ya's well known characterization of the space of Hardy-weights for the $p$-Laplacian. In addition, we provide various sufficient conditions on the potential $V$ and the Hardy-weight $g$ such that the best constant of the corresponding variational  problem is attained in an appropriate Beppo-Levi space.
 
This talk is based on a joint work with Ujjal Das.