Variable exponent modulus in symmetric domains
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914427598536704 |
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| author | Kargar, Rahim |
| author_facet | Kargar, Rahim |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26941 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Variable exponent modulus in symmetric domains Kargar, Rahim Complex Variables 46E35, 30C65, 31C15 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. |
| title | Variable exponent modulus in symmetric domains |
| topic | Complex Variables 46E35, 30C65, 31C15 |
| url | https://arxiv.org/abs/2603.26941 |