2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169115It is shown by a counterexample that isocapacitary and isoperimetric constants of a multi-dimensional Euclidean domain starshaped with respect to a ball are not equivalent. Sharp integral inequalities involving the harmonic capacity which imply Faber-Krahn property of the fundamental Dirichlet-Laplace eigenvalue are obtained. Necessary and sufficient conditions ensuring integral inequalities between a difference seminorm and the $L_p$-norm of the gradient are found.Functional AnalysisAnalysis of PDEs54C40, 14E20text