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Flow approach on the monotonicity of shape functionals

Yong Huang, Qinfeng Li, Shuangquan Xie, Hang Yang

TL;DR

This work develops a geometric-flow framework to study two classical shape functionals, the torsional rigidity $T(\Omega)$ and the first Dirichlet eigenvalue $\lambda_1(\Omega)$. By constructing deformation flows—notably height-based stretching/compression for polygons and mean curvature flow for smooth convex bodies—it establishes new monotonicity results and provides flow-based proofs of key inequalities, including a Saint-Venant-type inequality and rigidity statements for triangles, rhombuses, and rectangles. The authors also connect these results to continuous Steiner symmetrization, offering alternative proofs and insights while highlighting open questions about monotonicity along various curvature-driven flows. Overall, the paper demonstrates that a flow-based perspective can yield sharp, constructive comparisons among general shapes without relying on classical symmetrization techniques. The methods have potential to extend to nonlocal energies and other shape functionals, broadening the toolkit for shape optimization problems.

Abstract

We develop a geometric flow framework to investigate the following two classical shape functionals: the torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. First, by constructing novel deformation paths governed by stretching flows, we prove several new monotonicity properties of the torsional rigidity and the first eigenvalue along the evolutions on triangles and rhombuses. These results also lead to new and simpler proofs of some known results, without using the Steiner symmetrization argument. Second, utilizing the mean curvature flow, we give a new proof of the Saint-Venant inequality for smooth convex bodies. Third, by discovering a gradient norm inequality for the sides of rectangles, we prove monotonicity and rigidity results of the torsional rigidity on rectangles.

Flow approach on the monotonicity of shape functionals

TL;DR

This work develops a geometric-flow framework to study two classical shape functionals, the torsional rigidity and the first Dirichlet eigenvalue . By constructing deformation flows—notably height-based stretching/compression for polygons and mean curvature flow for smooth convex bodies—it establishes new monotonicity results and provides flow-based proofs of key inequalities, including a Saint-Venant-type inequality and rigidity statements for triangles, rhombuses, and rectangles. The authors also connect these results to continuous Steiner symmetrization, offering alternative proofs and insights while highlighting open questions about monotonicity along various curvature-driven flows. Overall, the paper demonstrates that a flow-based perspective can yield sharp, constructive comparisons among general shapes without relying on classical symmetrization techniques. The methods have potential to extend to nonlocal energies and other shape functionals, broadening the toolkit for shape optimization problems.

Abstract

We develop a geometric flow framework to investigate the following two classical shape functionals: the torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. First, by constructing novel deformation paths governed by stretching flows, we prove several new monotonicity properties of the torsional rigidity and the first eigenvalue along the evolutions on triangles and rhombuses. These results also lead to new and simpler proofs of some known results, without using the Steiner symmetrization argument. Second, utilizing the mean curvature flow, we give a new proof of the Saint-Venant inequality for smooth convex bodies. Third, by discovering a gradient norm inequality for the sides of rectangles, we prove monotonicity and rigidity results of the torsional rigidity on rectangles.

Paper Structure

This paper contains 11 sections, 18 theorems, 100 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Monotonicity along stretching flows
  3. A new proof of the Saint-Venant inequality by the mean curvature flow
  4. On averaged squared norm of the gradient of torsion function, rigidity and monotonicity results on rectangles
  5. Preliminaries
  6. Monotonicity under the shortest height stretching flow and tallest height compressing flow
  7. monotonicity along leg stretching flow
  8. A New proof of the Saint-Venant inequality via the mean curvature flow
  9. Monotonicity on rectangles via flow approach
  10. An alternative proof of Proposition \ref{['rectangle-thm']} via continuous symmetrization
  11. Further remarks on Question \ref{['kq1']}

Key Result

Theorem 1.3

Let $\Omega=\triangle_{ABC}$ be a triangle, with $AB$ being the longest side and lying on the $x$-axis, and with $C$ lying on the positive $y$-axis. Suppose that $|AB|>|BC|\ge |AC|$, and we let $C_t=(1+t)C$. Let $t_1>0$ be the unique number such that $|BA|=|BC_{t_1}|$. Then, for any $t\in (-1,t_1)$,

Figures (11)

  • Figure 1.1: Another illustration of Theorem \ref{['yangsheng2']} for the case of $t\in (0,t_1]$. Stretching the shortest height $OC$ above while scaling to keep the area fixed. $|\triangle_{A_tB_t\tilde{C}_t}|=|\triangle_{ABC}|$, $T(\triangle_{A_tB_t\tilde{C}_t})$ is strictly increasing, $\lambda_1(\triangle_{A_tB_t\tilde{C}_t})$ is strictly decreasing.
  • Figure 1.2: Another illustration of Theorem \ref{["yangsheng2'"]} for the case of $0<t\le t_2$. Compressing below the tallest height $OC$ while scaling to keep the area fixed. $|\triangle_{A_tB_t\tilde{C}_t}|=|\triangle_{ABC}|$, $T(\triangle_{A_tB_t\tilde{C}_t})$ is strictly increasing and $\lambda_1(\triangle_{A_tB_t\tilde{C}_t})$ is strictly decreasing.
  • Figure 3.1: For $-1<t<t_1$, $|AB|>|BC_t|\ge |AC_t|$. $y=x\tan \beta_t$ and $y=-x\tan \gamma_t$ pass through the midpoints of $BC_t$ and $AB_t$, respectively. $|p_tM_t|=|p_t'M_t|$, $|\nabla u(t)(p_t)|<|\nabla u(t)(p_t')|$, $|q_tN_t|=|q_t'N_t|$, $|\nabla u(t)(q_t)|<|\nabla u(t)(q_t')|$.
  • Figure 3.2: Stretch the diagonal $BD$ of the square $\square ABCD$. During the stretching process, $T(\cdot)/|\cdot|^2$ is increasing, while $\lambda_1(\cdot)|\cdot|$ is decreasing.
  • Figure 4.1: Stretching one leg and scaling to fix the area
  • ...and 6 more figures

Theorems & Definitions (36)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Proposition 1.9
  • Theorem 1.10
  • Proposition 1.11
  • Theorem 2.1: Pohozaev's identity.
  • ...and 26 more