Isocapacitary constants for the $p$-Laplacian on compact manifolds

In this paper, we introduce Steklov and Neumann isocapacitary constants for the $p$-Laplacian on compact manifolds. These constants yield two-sided bounds for the $(p,\alpha )$-Sobolev constants, which degenerate to upper and lower bounds for the first nontrivial Steklov and Neumann eigenvalues of the $p$-Laplacian when $\alpha =1$.