Weighted Dirichlet-type inequalities for the decreasing rearrangement in cylinders

Abstract: In this paper weighted Dirichlet-type inequalities for the decreasing rearrangement in cylinders are proved. A weighted isoperimetric inequality is also obtained.

• The paper develops novel weighted Dirichlet-type inequalities that generalize classical Pólya–Szegő results for cylindrical domains.
• It employs convexity, layer-cake representation, and piecewise linear approximations to extend inequalities to weighted Sobolev and BV spaces.
• Results provide sharp energy reduction under fiberwise decreasing rearrangement, aiding analysis in anisotropic and axially symmetric PDEs.

Weighted Dirichlet-type Inequalities for Decreasing Rearrangement in Cylinders

Introduction and Motivation

The paper addresses the development and analysis of weighted Dirichlet-type inequalities tied to the decreasing rearrangement of functions, with a focus on cylindrical domains. The classical rearrangement inequalities and symmetrization methods (e.g., Schwarz and Steiner symmetrization) underpin many results in the theory of variational problems, especially regarding minimizers' symmetry, comparison principles, and a priori estimates. However, classical symmetrizations become suboptimal when the problem's geometry is anisotropic or when weights are present in the underlying functionals.

Recent interest has emerged in establishing sharp inequalities for functionals in weighted settings or with respect to non-standard measures (e.g., Gaussian, monomial weights). The present work generalizes classical Pólya–Szegő-type inequalities, introducing a framework for weighted decreasing rearrangement in one direction, naturally suiting analysis in cylindrical domains or problems with preferred structural axes.

One-dimensional Weighted Rearrangement Inequalities

The authors begin by considering functions defined on intervals, introducing the class of admissible weight functions $f$ subject to continuity, positivity, and structural conditions ensuring subadditivity or symmetry. For a nonnegative function $u$ and its decreasing rearrangement $u^*$, they establish the following general Dirichlet-type inequality for suitable integrands $G$: $$\int_I G(f|u'|, u)\, dx \geq \int_I G(f|(u^*)'|, u^*)\, dx.$$ Crucially, $G$ must be continuous, convex, and monotone in the gradient argument, covering a range of applications including the $L^p$-Sobolev norms as special cases. The main technical contribution consists of extending classical results (notably, Landes' inequality) to the weighted and rearrangement context. The proof employs density arguments via "nice" piecewise linear functions, exploitation of convexity, and careful combinatorial analysis of rearrangement layers.

A weighted isoperimetric inequality for subsets $M \subset I$ is also derived, showing that their perimeter with respect to $f$ does not decrease under decreasing rearrangement. This structural result provides a foundation for subsequent functional inequalities.

Multidimensional Setting: Cylindrical Domains

The paper then generalizes these results to functions $u$ on cylindrical domains $u$0, with $u$1 a bounded Lipschitz domain in $u$2. The decreasing rearrangement $u$3 is defined with respect to the distinguished direction $u$4, rearranging each fiber independently. The main theorem extends the one-dimensional inequality: $u$5 where $u$6 is convex and monotone in the vertical derivative, and $u$7 serves as the weight. Through careful layer-cake representation, the proof constructs appropriate inverse functions and exploits Jensen's inequality in conjunction with the combinatorial properties of the rearrangement.

The central corollary, for $u$8 and $u$9, yields: $u^*$0 establishing that the weighted Dirichlet integral is non-increasing under fiberwise decreasing rearrangement.

Extension to Weighted Spaces: Sobolev and BV Inequalities

Recognizing the prevalence of weighted Sobolev and BV spaces in applications—including those with measures of the form $u^*$1—the authors introduce a systematic framework for rearrangement with respect to a general weight $u^*$2. This is formalized via the $u^*$3-rearrangement, defined by a change of variable involving the weight's primitive function.

For $u^*$4, it is shown that the rearranged function $u^*$5 maintains or decreases the $u^*$6-weighted norm: $u^*$7 Similarly, a corresponding result holds for the weighted BV norm.

An isoperimetric inequality for the $u^*$8-perimeter is established: For any measurable set $u^*$9, its $G$0-perimeter does not increase under $G$1-rearrangement,

$G$2

Analytical and Applied Implications

These inequalities capture the minimality of the Dirichlet-type energy under fiberwise monotonic rearrangement for a wide class of cylindrical geometries and weighted measures. On an analytical level, the results provide:

• A toolkit for sharp estimates in weighted variational problems.
• Generalizations of the classical Pólya–Szegő principle to settings where anisotropy or weights are present.
• Structural results that can be used to deduce symmetry, uniqueness, or stability properties of minimizers in PDEs with axial or weighted structure.

Applied implications arise in multiple domains:

• Weighted inequalities are central in generalized axially symmetric potential theory, which reduces higher-dimensional PDEs to non-uniformly elliptic equations in geometrically reduced spaces.
• Applications in fluid mechanics (axially symmetric flows), torsion problems for shafts, and the study of virtual mass and polarization for bodies of revolution benefit from this framework, as such problems are governed by differential equations and variational principles with inherent anisotropy and weightings.

Future Directions

Given the general structure of the established inequalities, there are several directions for further research:

• Extension to fully anisotropic or tensor-valued weights, incorporating more complex geometric dependencies.
• Exploration of equality conditions and stability bounds to characterize when minimization via rearrangement is strict or degenerate.
• Application to nonlocal operators or fractional Sobolev spaces, where the fiberwise rearrangement could interact non-trivially with the global nature of the energy functional.
• Investigation of analogous principles for nonlinear or degenerate elliptic equations, especially in the presence of mixed boundary conditions or inhomogeneous measures.

Conclusion

The paper systematically extends Dirichlet-type inequalities to weighted, fiberwise decreasing rearrangements within cylindrical domains, unifying and generalizing several frameworks for handling anisotropy and weighted measures in variational analysis. The presented results establish new weighted isoperimetric inequalities, reinforce the utility of rearrangement techniques in PDEs and geometric analysis, and open further avenues for applying such inequalities in both mathematical theory and practical modeling scenarios.