Anisotropic gradient rearrangement of BV functions and applications

Technical Summary: Anisotropic Gradient Rearrangement of BV Functions and Applications

Problem Statement
The paper addresses the extension of symmetrization and rearrangement techniques from the Euclidean setting to anisotropic environments for functions of bounded variation (BV). While classical symmetrization (e.g., Schwarz symmetrization) compares solutions of Partial Differential Equations (PDEs) on general domains to solutions on symmetric balls, recent developments in [10] introduced a gradient rearrangement for BV functions in the Euclidean case. This procedure separates the absolutely continuous part of the gradient from its singular part, rearranging the former while concentrating the latter on the boundary of the symmetrized domain.

The specific problem tackled here is the generalization of this result to the anisotropic setting, where the geometry is governed by a convex, positively 1-homogeneous function $H$ (the anisotropic norm or Finsler norm) rather than the Euclidean norm. In this context, the natural geometric objects are Wulff shapes (minimizers of anisotropic perimeter) rather than Euclidean balls. The challenge lies in defining a rearrangement that preserves the anisotropic total variation and establishes an $L^1$ comparison between a function and its symmetrized counterpart, accounting for the singular part of the distributional derivative.

Methodology
The authors employ a strategy analogous to the Euclidean case in [10], adapted to the anisotropic framework:

1. Definitions and Preliminaries: The paper establishes the necessary notation for anisotropic norms ($H$), polar norms ($H^\circ$), and Wulff balls ($W_R$). It defines the anisotropic total variation $|Du|_H$ and recalls the anisotropic Coarea formula and isoperimetric inequality.
2. Truncation and Measure Decomposition: For a non-negative function $u \in BV_0(\Omega)$, the authors consider truncations $v_s = (u - u^*(s))_+$. They analyze the anisotropic total variation of these truncations, $G(s) = |Dv_s|_H(\mathbb{R}^n)$, decomposing it into an absolutely continuous part ($G_{H,1}$) and a singular part ($G_{H,2}$).
3. Gradient Rearrangement Construction:
   • The absolutely continuous part of the anisotropic energy density, $H(\nabla_a u)$, is rearranged to an increasing function $g^\sharp$ on the Wulff ball $\Omega^\star$.
   • A new function $u^\star$ is constructed on $\Omega^\star$ such that its gradient satisfies $H(\nabla u^\star) = g^\sharp$ almost everywhere.
   • Crucially, the singular part of the original derivative, $D_s u$, is not discarded. Instead, it is encoded as a constant boundary trace $c_0$ on $\partial \Omega^\star$. This constant is defined as the ratio of the total singular anisotropic variation to the anisotropic perimeter of the Wulff ball: $c_0 = |D_s u|_H(\mathbb{R}^n) / P_H(\Omega^\star)$.
4. Comparison Proof: The main theorem is proved by establishing a differential inequality relating the distribution function of $u$ to the measure induced by the rearrangement. By integrating this inequality and utilizing the Hardy-Littlewood inequality, the authors derive the $L^1$ comparison.

Key Contributions and Results

• Theorem 1.2 (Main Result): For a non-negative function $u \in BV_0(\Omega)$, let $u^\star$ be the anisotropic gradient rearrangement defined on the Wulff ball $\Omega^\star$ (with $|\Omega^\star| = |\Omega|$). The paper proves the $L^1$ comparison:
  $$\|u\|_{L^1(\Omega)} \leq \|u^\star\|_{L^1(\Omega^\star)}$$
  This result generalizes the Euclidean $L^1$ comparison from [10] to the anisotropic setting.

• Preservation of Anisotropic Total Variation: The construction ensures that the total anisotropic variation is preserved under symmetrization:
  $$|Du^\star|_H(\mathbb{R}^n) = |Du|_H(\mathbb{R}^n)$$
  The singular part of the derivative is explicitly recovered as a boundary term in the symmetrized function.

• Applications to Torsional Rigidity: The authors apply the main theorem to derive Saint-Venant type inequalities for two functionals related to torsional rigidity in the anisotropic setting:

  1. Penalized Torsional Rigidity ($T^H_F$): For a functional involving anisotropic Dirichlet energy and a penalization term based on the measure of the support of the gradient, the paper proves that the functional is minimized (or the rigidity maximized) when the domain is a Wulff ball. This is achieved by showing that the anisotropic gradient rearrangement preserves the energy and the penalization term while increasing the $L^1$ norm of the function.
  2. Anisotropic Insulation Functional ($T^H_G$): For a functional involving anisotropic Dirichlet energy and a boundary term related to the trace of the function, the paper establishes a similar Saint-Venant inequality. A key technical step (Lemma 4.4) identifies the boundary integral of the trace with the singular part of the anisotropic variation of the zero extension of the function, allowing the main rearrangement theorem to apply.

Significance and Claims
The paper claims to provide a rigorous anisotropic counterpart to the gradient symmetrization technique introduced in [10]. Its primary significance lies in:

• Handling Singularities: It successfully extends rearrangement methods to BV functions by explicitly tracking the singular part of the derivative through a boundary datum, rather than losing it during symmetrization.
• Geometric Generality: It replaces the Euclidean ball with the Wulff shape, which is the natural minimizer for anisotropic surface energies, thereby making the results applicable to models involving crystalline surface tension and interfacial energies.
• Variational Applications: It demonstrates the utility of this rearrangement in proving isoperimetric-type inequalities for variational problems where the natural geometry is anisotropic.

Limitations and Modesty
The authors are modest regarding the scope of their uniqueness claims. In the context of the penalized torsional rigidity on a Wulff ball (Proposition 4.1), they establish the existence of a symmetric minimizer but explicitly state that they do not claim uniqueness. They note that a uniqueness or rigidity result would require an anisotropic analogue of the rigidity theorems proved in [33] for the Euclidean case, which are not available in the current anisotropic setting. Consequently, the paper focuses on the comparison principles and existence of symmetric minimizers rather than full rigidity characterizations.