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Variable exponent modulus in symmetric domains

Rahim Kargar

Abstract

We develop explicit variational formulas for the $p(\cdot)$-modulus of curve families in symmetric domains of $\mathbb{R}^n$, under a log-Hölder continuous exponent $p\colonΩ\to(1,\infty)$, where $Ω$ 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.

Variable exponent modulus in symmetric domains

Abstract

We develop explicit variational formulas for the -modulus of curve families in symmetric domains of , under a log-Hölder continuous exponent , where 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 -quasiconformal mappings distort the -modulus and capacity by controlled factors. Applications and numerical examples are included.

Paper Structure

This paper contains 26 sections, 17 theorems, 136 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on Variable Exponent Spaces
  3. Log-Hölder Continuity
  4. Variable Exponent Lebesgue Spaces
  5. Curve Families, Modulus, and Capacity
  6. Standard Tools
  7. Motivating Examples and Model Geometries
  8. Variational Characterization and Reduction to One Dimension
  9. Euler--Lagrange Characterization
  10. Reduction to Radial Densities for the Annulus
  11. Reduction to a One-Dimensional Problem
  12. Explicit Formula for the Extremal Density
  13. Test Densities and Upper Bounds for Annuli and Cylinders
  14. Modulus of a Cylindrical Domain
  15. Upper Bound for the Cylinder
  16. ...and 11 more sections

Key Result

Lemma 2.2

Let $(\Sigma,\mu)$ be a probability space, $\phi\colon\mathbb{R}\to\mathbb{R}$ convex, and $f\in L^1(\Sigma,\mu)$. Then In particular, let $D\subset\mathbb{R}^{n-1}$ be measurable with $0<|D|<\infty$, and let $f\ge 0$ be measurable. Then for any $q\ge 1$,

Theorems & Definitions (40)

  • Definition 2.1: See Diening2011
  • Lemma 2.2: See Diening2011
  • Definition 2.3: See HHM
  • Definition 2.4: See AK04HL
  • Lemma 2.5: See EGF59
  • Lemma 2.6: See Diening2011
  • Example 3.1
  • Example 3.2
  • Theorem 4.1
  • proof
  • ...and 30 more