Flux-Hardy Inequalities with Optimal Constants via Divergence and Rearrangements

Abstract

We develop sharp Hardy-type inequalities for boundary flux functionals generated by dilations of a fixed smooth set in R^n. The method combines the divergence theorem with rearrangement bounds to reduce flux estimates to one-dimensional Hardy averages. Directional, divergence-form, and multi-flux inequalities are obtained in arbitrary dimension with the optimal constant and sharpness is shown by explicit near-extremizing constructions.