Variable exponent modulus in symmetric domains

TL;DR

This paper derives explicit formulas for the variable exponent p-modulus of curve families in symmetric domains, using symmetrization techniques and variational analysis, with applications to quasiconformal mappings.

Contribution

It introduces explicit variational formulas for the p-modulus in symmetric domains with variable exponents, extending classical results and providing tools for distortion analysis.

Findings

Explicit formulas for the p-modulus and extremal densities in symmetric domains.

A duality between capacity and modulus is established.

Controlled distortion bounds for quasiconformal mappings are proved.

Abstract

We develop explicit variational formulas for the $p(\cdot)$-modulus of curve families in symmetric domains of $\mathbb{R}^n$, under a log-H\"older continuous exponent $p\colon\Omega\to(1,\infty)$, where $\Omega$ is an open set. For annuli with radial exponent and cylinders with axial exponent, spherical symmetrization and averaging over transverse variables reduce the problem to a one-dimensional variational problem. The extremal density is uniquely characterized by a pointwise Euler--Lagrange condition with a Lagrange multiplier determined by a normalization constraint, yielding explicit formulas for both the density and the modulus. We also establish a two-sided capacity--modulus duality and prove that $K$-quasiconformal mappings distort the $p(\cdot)$-modulus and capacity by controlled factors. Applications and numerical examples are included.