2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119235Motivared by Carleman's proof of the isoperimetric inequality in the plane, we study some sharp integral inequalities for harmonic functions on the upper halfspace. We also derive the regularity for nonnegative solutions of the associated integral system and some Liouville type theorems.35 pagesAnalysis of PDEsFunctional Analysis26D10, 45Mxxtext