Shape optimization under width constraint: the Cheeger constant and the torsional rigidity

Abstract

In this article it is shown that the equilateral triangle maximizes the Cheeger constant and minimizes the torsional rigidity among shapes having a fixed minimal width. The proof techniques use direct comparisons with simpler shapes, consisting of disks with three disjoint caps. Comparison results for harmonic functions help establish that in non-equilateral configurations the shape derivative has an appropriate sign, contradicting optimality.

Figures (7)

• Figure 1: (left) Existence of a three-$r$-cap subset for a shape with given minimal width. (right) Example of a three-$r$-cap shape: convex hull of a disk $D_r$ of center $O$ and radius $r \in [1/3,1/2]$ and points $A,B,C$ such that $AO=BO=CO=1-r$.
• Figure 2: A three-$r$-cap shape $T_{ABC}$ together with the arcs $\gamma_A, \gamma_B,\gamma_C$ near points $A,B,C$. In the figure, when $AB<AC$, the symmetric of the left part with respect to the vertical axis is contained in the right part.
• Figure 3: Illustration for boundary conditions for $u_\theta, u_{\theta'}$ given by \ref{['eq:harmonic-slide']} for $\theta<\theta'$. Dirichlet boundary conditions $u_\theta=0$ and $u_{\theta'}=0$ hold everywhere on the boundary of the half-disk (including the diameter), except the arcs $\gamma_\gamma$ and $\gamma_{\theta'}$, respectively. On these arcs we have a boundary condition given by the same non-negative function $\phi$.
• Figure 4: Geometric configuration for problem \ref{['eq:harmonic-K']} on the domain $K$, the convex hull of the half disk and a point $A$ on the real axis. The boundary condition is $u_\theta=0$ except an arc $\gamma_\theta$ which slides towards $A'$. Values of $u_\theta$ decrease in $\mathcal{C}_A$ as the positive boundary condition slides.
• Figure 5: A three-$r$-cap shape with $AB<AC$ such that $A$ is on the positive real axis. The vertex $C$ is symmetrized with respect to the $x$-axis. The difference of the torsion function and its symmetrized will be compared in $\text{conv}(D_+,A)$.

Theorems & Definitions (16)

• Theorem 1
• Definition 2
• Theorem 3
• Conjecture 4
• Theorem 5
• Proposition 6
• Proposition 7
• Proposition 8
• Proposition 9
• Proposition 10