Variable exponent modulus in symmetric domains

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 $p\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\to (1,\mathrm{\infty })$, where $\mathrm{\Omega }$ 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.