The equivalences among p-capacity, p-Laplace-capacities and Hausdorff measure

Let Ω be a smooth bounded domain of R . In this paper, we study the equivalences among three capacities and Hausdorff measure. First we present the equivalence between p-capacity Cp(K) and p-Laplace-capacitiy C∆p(K) relative to Ω for any compact set K ⊂ Ω. Secondly we establish the equivalence between p-Laplace capacity Cp(K, ∂Ω) relative to ∂Ω and Hausdorff measure HN−1(K) for any compact set K ⊂ ∂Ω.