Hardy-type inequalities derived from [formula omitted]-harmonic problems

Abstract

We consider the anti-coercive partial differential inequality of elliptic type involving p-Laplacian: −Δpu≥Φ, where Φ is a given locally integrable function and u is defined on an open subset Ω⊆Rn. Knowing solutions, we derive Caccioppoli inequalities for u. As a direct consequence we obtain Hardy inequalities involving certain measures for compactly supported Lipschitz functions. Our methods allow to retrieve classical Hardy inequalities with optimal constants. We present several applications leading to various weighted Hardy inequalities.