A flow approach to the monotonicity of shape functionals

TL;DR

This paper introduces a geometric flow framework to analyze and prove monotonicity properties of classical shape functionals like torsional rigidity and the first Laplacian eigenvalue, offering new proofs and insights.

Contribution

It develops novel deformation flows and a mean curvature flow approach to establish monotonicity and rigidity results, simplifying proofs and proposing new conjectures.

Findings

Monotonicity of shape functionals under specific flows for triangles and rhombuses

A new geometric proof of the Saint-Venant inequality for convex domains

Monotonicity and rigidity results for torsional rigidity on rectangles

Abstract

We develop a geometric flow framework to investigate two classical shape functionals: the torsional rigidity and the first Dirichlet eigenvalue of the Laplacian. First, by constructing novel deformation paths governed by height-stretching flows, leg-stretching flows, and angle-bisector flows, we prove new monotonicity properties for these functionals under deformations of triangles and rhombuses. These results also lead to new and simpler proofs of some known results, without using the Steiner symmetrization argument. Second, we introduce a mean curvature flow approach to the Saint-Venant inequality, providing a new geometric proof for smooth convex domains. We establish a weak monotonicity property along the flow and characterize the equality case, which leads to the discovery of an intriguing new functional whose extremal properties suggest a further conjecture. Third, by discovering a gradient norm inequality for the sides of rectangles, we prove monotonicity and rigidity results of the torsional rigidity on rectangles.