$p$-variational capacity of interior condensers and geometric reduction by a fixed phase

Abstract

We study the $p$-variational capacity of interior condensers in a bounded open set $\Omega\subset\mathbb R^n$ when both plates are determined by a single phase $\theta:\Omega\to\mathbb R$ in $W^{1,\infty}(\Omega)$ through sublevel and superlevel sets. By restricting the admissible class to potentials of the form $u=v\circ\theta$ and applying the coarea formula, the problem reduces to a one-dimensional variational functional in the level variable, governed by an \textit{energy weight} that combines the gradient profile of $\theta$ and the geometry of its level sets. We obtain an explicit formula for the \textit{reduced problem}, construct an explicit optimal profile, and deduce an upper bound for the full geometric capacity by fibered restriction. In addition, we derive estimates for the energy weight in terms of the gradient and the size of the fibers, and analyze the local effect of critical levels on the integrability of the reduced resistance. Finally, we present symmetric models in which the fibered reduction coincides with the full geometric capacity and, in the linear case, a quantitative tangential obstruction to that exactness.