Variable exponent modulus in symmetric domains

Abstract

We develop a rigorous variational theory for the modular variable-exponent modulus of curve families in two symmetric geometries: annuli $A(r_1,r_2)\subset {\mathbb{R}}^n$ with radial exponent $p(x)=q_0(|x|)$, and cylinders $\mathcal{C}=D\times (0,L)$ with axial exponent $p(x^{\prime },t)=\eta (t)$, under the assumption $1<p^{-}\le p^{+}<\infty$. For annuli, spherical averaging reduces the modulus problem to a one-dimensional constrained variational problem and yields a unique positive extremal density \begin{equation*} ρ_*(r) = \left( \fracλ {ω_{n-1}q_0(r)r^{n-1}} \right)^{1/(q_0(r)-1)}, \end{equation*} where $\lambda >0$ is uniquely determined by the normalization ${\int }_{r_1}^{r_2}{\rho }_{\ast }(r)dr=1$. An analogous fiber-averaging argument reduces the cylindrical problem to one dimension and gives the corresponding unique extremal profile. We recover the classical logarithmic and constant extremal densities in the conformal and constant-exponent cases, respectively, and derive explicit upper bounds from these test densities. We further establish equality between the relative variable-exponent capacity of compact condensers and the corresponding curve modulus under the stated regularity assumptions, obtaining explicit capacity formulas in the radially symmetric annular setting. Finally, we discuss the obstruction to quasiconformal quasi-invariance for nonconstant exponents and formulate a related open problem, and illustrate the variational formulas and test-density bounds through numerical examples.

35 pages