Geometric properties of optimizers for the maximum gradient of the torsion function

TL;DR

The paper proves that, within planar convex domains, the maximal gradient of the torsion function (normalized by area or perimeter) is attained by an optimizer whose boundary is C^1 and which contains a boundary line segment where the gradient reaches its maximum. The authors develop a framework combining a boundary Harnack principle for the torsion function, probabilistic representations, and Blaschke compactness to establish existence, regularity (no corners), and the line-segment structure. A perturbation argument shows that if the boundary lacked such a segment, one could construct a competitor with a strictly larger objective, contradicting optimality. The results extend to the perimeter-normalized functional J_P, using analogous arguments and careful perimeter accounting. This provides a rigorous geometric picture of optimal shapes for maximal gradient of the torsion function in convex planar domains.

Abstract

Consider $J(Ω):= \|\nabla u_Ω\|_\infty/\sqrt{|Ω|} $ and $J_P(Ω):= \|\nabla u_Ω\|_\infty/P(Ω) $, where $Ω$ is a planar convex domain, $u_Ω$ is the torsion function, $P(Ω)$ is the perimeter of $Ω$ and $|Ω|$ its area. We prove that there exist planar convex domains that maximize the functionals $J$ and $J_P$, and any maximizer has a $C^1$ boundary that contains a line segment on which $|\nabla u_Ω|$ attains its maximum.

Figures (12)

• Figure 1: Subdomains of $\Omega$. Drawing not to scale.
• Figure 2: The domain $\Omega$ (red dashed line) and the shifted domain $\Omega'_n$ (black).
• Figure 3: A detailed version of Fig. \ref{['fig:shift_v_n']}.
• Figure 4: The domain $A_n$ and relevant notations.
• Figure 5: A domain $\Omega$ with corners.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Lemma 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof
• Proposition 4.1
• proof