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arxiv: 2604.11448 · v1 · submitted 2026-04-13 · 🧮 math.AP · math.FA

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

Vicente Vergara This is my paper

Pith reviewed 2026-05-10 15:35 UTC · model grok-4.3

classification 🧮 math.AP math.FA
keywords p-variational capacityinterior condensersphase functioncoarea formulafibered reductionenergy weightsymmetric modelstangential obstruction
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The pith

Restricting test functions to compositions with a fixed phase function reduces the p-variational capacity of an interior condenser to an explicit one-dimensional weighted integral.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that when the two plates of an interior condenser in a domain are the sublevel and superlevel sets of a single Lipschitz phase function θ, the p-capacity can be bounded from above by restricting admissible potentials to the special form v composed with θ. Applying the coarea formula then collapses the original minimization into a one-dimensional variational problem whose integrand is an energy weight built from the gradient of θ and the surface measures of its level sets. This yields an explicit formula, an optimal profile that can be written down by hand, and a computable upper bound; the bound is sharp precisely when the geometry is symmetric with respect to the phase.

Core claim

By restricting the admissible class to potentials of the form u = v ∘ θ and applying the coarea formula, the p-variational capacity problem reduces to a one-dimensional variational functional in the level variable whose energy weight combines the gradient profile of θ and the geometry of its level sets. An explicit formula for the reduced problem is obtained, an optimal profile is constructed, and an upper bound for the full geometric capacity follows by fibered restriction. The reduction is exact in symmetric models, while a quantitative tangential obstruction appears in the linear case.

What carries the argument

The fibered restriction to potentials of the form v ∘ θ together with the coarea formula, which produces a weighted one-dimensional energy functional whose weight encodes both |∇θ| and the (n-1)-dimensional measure of the level sets.

If this is right

  • The reduced one-dimensional problem supplies an explicit upper bound on the p-capacity that can be evaluated once the energy weight is known.
  • An optimal profile for the reduced functional immediately yields a concrete test function for the original capacity problem.
  • Estimates relating the energy weight to |∇θ| and the size of level sets give integrability criteria near critical levels.
  • In symmetric models the reduced value coincides with the actual capacity, so the method solves the problem exactly.
  • The quantitative tangential obstruction in the linear case measures how far the bound deviates from equality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same fibered restriction could be applied to other variational integrals whose integrands are invariant under level-set foliations, turning them into weighted one-dimensional problems.
  • The gap between the reduced bound and the true capacity in non-symmetric cases might be controlled by integral curvature terms involving the second fundamental form of the level sets.
  • Numerical approximation schemes for capacity could first solve the cheap one-dimensional reduced problem and then use the resulting profile as a warm start for the full-dimensional minimization.

Load-bearing premise

That the infimum over the restricted class of functions v ∘ θ is sufficiently close to the true infimum to produce a useful upper bound on the capacity.

What would settle it

An explicit non-symmetric condenser whose true p-capacity, computed by other means, exceeds the value of the reduced one-dimensional problem by an amount larger than the tangential-obstruction term identified in the linear case.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript studies the p-variational capacity of interior condensers in a bounded open set Ω ⊂ ℝ^n whose plates are the sublevel and superlevel sets of a fixed phase θ ∈ W^{1,∞}(Ω). By restricting test functions to the subclass u = v ∘ θ and applying the coarea formula, the capacity problem reduces to a one-dimensional variational problem in the level variable whose energy weight combines |∇θ| with the (n-1)-dimensional geometry of the level sets of θ. The authors derive an explicit formula for this reduced functional, construct the optimal profile v explicitly, obtain an upper bound on the original geometric capacity, supply estimates for the weight in terms of the gradient and fiber sizes, analyze integrability issues at critical levels of θ, and identify symmetric models in which the fibered reduction is exact together with a quantitative tangential obstruction in the linear case.

Significance. If the reduction holds, the work supplies a constructive geometric method for producing explicit upper bounds on p-capacities of level-set condensers and for identifying the precise geometries in which the bound is sharp. The explicit optimal profile, the estimates on the energy weight, and the integrability analysis at critical levels are concrete strengths that make the technique applicable in potential theory and geometric PDEs. The characterization of equality cases and the linear obstruction further clarify the scope of the reduction.

minor comments (2)
  1. The definition of the energy weight (introduced after the coarea reduction) would benefit from an immediate displayed formula that isolates the contributions of |∇θ| and the level-set measure; this would improve readability when the weight is later estimated.
  2. In the discussion of critical levels, a brief remark on the precise Sobolev regularity needed for the coarea formula to yield an integrable weight would help readers verify the integrability claims without consulting external references.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending acceptance. We appreciate the detailed summary of the main results and the recognition of the explicit constructions, estimates, and analysis of equality cases.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation restricts admissible potentials to the subclass u = v ∘ θ, applies the coarea formula (standard for W^{1,∞} functions) to obtain a weighted 1D variational problem, constructs an explicit optimizer for the reduced functional, and thereby obtains a valid upper bound on the true capacity (since inf over subclass ≥ true inf). This is a direct variational inequality with no reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. Symmetric models where equality holds are identified separately by direct verification, and the tangential obstruction in the linear case is derived from explicit computation. The argument is self-contained and relies only on standard tools (coarea, direct method in calculus of variations) without circular dependence on its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard tools from Sobolev spaces and geometric measure theory without introducing new free parameters or postulated entities.

axioms (2)
  • standard math The coarea formula holds for functions in W^{1,∞}(Ω)
    Invoked to reduce the n-dimensional integral to an integral over level sets parameterized by the level variable.
  • standard math The p-variational capacity is defined as the infimum of the p-energy over admissible functions separating the plates
    Standard definition in the field used to set up the problem and the admissible class.

pith-pipeline@v0.9.0 · 5499 in / 1517 out tokens · 89465 ms · 2026-05-10T15:35:49.594299+00:00 · methodology

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