1. Eigenfunctions
Open questions about the shape and properties of ground states and torsion functions.-
$L^1$ norm of first eigenfunction.
Conjecture 1.1.
[Phanuel Mariano] Let $\Omega\subset \mathbb{R}^n$ be an open set with finite measure, $\lambda_1(\Omega)$ be the first eigenvalue of the Laplacian with Dirichlet boundary conditions, and let \[T(\Omega)=\int_{\Omega}u\] where $u:\Omega\to \mathbb{R}$ is the torsion function given by the equation \[\begin{cases} -\Delta u=1 & \text{ in }\Omega\\ u=0 & \text{ in }\partial\Omega \end{cases}\] Let $F(\Omega) = \lambda_1(\Omega)T(\Omega)/|\Omega|$. Then it is known that $F(\Omega)$ is bounded between $0$ and $1$, and these bounds are sharp. However when one restricts to convex subsets of $\mathbb{R}^2$, there is a better bound \[\frac{\pi^2}{32}\leq F(\Omega)\leq 0.9967\] The left-hand side is due to [Brasco, Mazzoleni], and the right-hand side to [Ftouhi]. The conjecture raised by Phanuel Mariano is as follows: for any planar convex set $\Omega$, does one has \[\frac{\pi^2}{24}\leq F(\Omega)\leq \frac{\pi^2}{12}\] If these bounds were verified, they would be sharp in the following sense: the right one is asymptotically reached for thin rectangles, while the left one is asymptotically reached for any sequence of triangles that collapses on the interval. More recently in [Banuelos, Mariano], these bounds were proven in the case where $\Omega$ is either a triangle or a rectangle. -
Maximal gradient of the torsion
Problem 1.2.
[Krzysztof Burdzy] This is a report of a question that was raised in (Hoskins, Steinerberger - 2021), and the following statement is based on this paper.
Let $\Omega \subset \mathbb{R}^2$ be a convex set, and $u_\Omega:\Omega\to \mathbb{R}$ be the torsion function defined by
\begin{align*} - \Delta u_\Omega = 1 , & \qquad \text{in $\Omega$,} \\ u_\Omega=0 , & \qquad \text{on $\partial \Omega$.} \end{align*}
It is known that $\lVert \nabla u \rVert_{L^\infty(\Omega)} \leq c |\Omega|^{1/2}$ for some constant $c<\frac{1}{\sqrt{2\pi}}\approx 0.398$, that cannot be taken smaller than $0.358$ (which is the best constant that is obtained numerically).
One could also obtain a slightly worse, explicit bound by looking at ellipses which is one of the few example with explicit torsion function $u_\Omega$).
The question raised in (Hoskins, Steinerberger -2021) is the following: for a convex set of $\mathbb{R}^2$ of given measure, how large can the gradient of the torsion function get ? In other words, what is \[\sup\left\{\Vert \nabla u_\Omega\Vert_{L^\infty(\Omega)},\ \Omega\subset \mathbb{R}^2\text{ convex s.t. }|\Omega|=1\right\}\ ?\]
Numerically, it is known that the disk is not optimal (even among ellipses), and the optimal set obtained numerically seem to have some flat portion on the boundary.
A particular point of interest is the location of the point of an optimal set where the maximal gradient is reached: is it possible to prove that such a point is necessarily on a flat part of the boundary in a well-quantified way ? -
Shape of the ground state
Problem 1.3.
[David Jerison] Let $\Omega\subset\mathbb{R}^n$ be a bounded convex set, and let $u$ be the lowest eigenfunction (ground state) of the Laplacian with Dirichet boundary conditions, with unit $L^2$ norm. A general question is to relate the domain $\Omega$ and the ground state $u$ with a rectangular domain $R$ associated to a ground state with separated variables.
For this we introduce the following notations: let $\xi \in \mathbb{R}^n$ be a unit vector ($|\xi|=1$). Let \[ P(\xi)=\int_{\partial \Omega} |\xi \cdot \nabla u|^2 \, dS , \qquad Q(\xi)=\int_\Omega |\xi \cdot \nabla u|^2 \, dx . \] $Q$ is the projection body function for volume.
It is known that \[ Q(\Omega) \leq \frac{c}{\text{inradius}(\Omega)^2} . \]
A first question is the relation between $P$ and $Q$: the first is linked to the shape derivative of $\lambda_1(\Omega)$ in the direction of $\xi$, while the second is related to the distribution of $\nabla u $ in $\Omega$.
A second question is whether $\log(u)$ is “essentially quadratic” on the interior of $\Omega$ in the following sense: it is known that $\log(u)$ is concave and reaches a unique maximum at some point $p^*$, the conjecture raised by David Jerison is whether there exists some constant $C_n>0$ such that for any $p\in \{u>\frac{1}{2}u(p^*)\}$, we have \[\frac{1}{C_n}\left(-\nabla^2 \log\ u(p)\right)\leq \left(-\nabla^2\log\ u(p^*)\right)\leq C_n\left(-\nabla^2\log\ u(p)\right).\]
Cite this as: AimPL: Symmetry-breaking of optimal shapes, available at http://aimpl.org/symmetrybreaking.