
## 2. The Dirichlet Laplacian

1. ### Multiplicity of optimal Dirichlet eigenvalues

#### Conjecture 2.1.

[A. Henrot] Let $\Omega_k^*$ be a minimizer of the shape optimization problem $$\min\{ \lambda_k(\Omega): \Omega\subset \mathbb{R}^n, |\Omega|=V_0\}.$$ Prove that $\lambda_{k-1}(\Omega_k^*)=\lambda_k(\Omega_k^*)$.
This conjecture is verified by E. Oudet in [MR2084326] via numerical experiments.
• ### Mahler inequality for the principal Dirichlet eigenvalue

Let $\mathcal{K}^n_\star$ be the class of centrally symmetric bounded convex domains in $\mathbb{R}^n$. For a domain $K\in\mathcal{K}^n_\star$ we define its polar set by $$K^\circ := \{x\in \mathbb{R}^n\colon x \cdot y \leq 1 \mbox{ for all }y \in K\}.$$ Further, let $\mathsf{GL}_n$ be the family of invertible linear transformation in $\mathbb{R}^n$.

#### Conjecture 2.2.

[E. Harrell] A hypercube in $\mathbb{R}^n$ is an attainer for the maximization problem $\sup\left\{ \inf_{T\in\mathsf{GL}_n} \lambda_1(T(K))\lambda_1((T(K))^\circ)\colon K\in\mathcal{K}^n_\star\right\}.$
According to Proposition 5 in [MR3551851], a maximizer for the above problem exists. Moreover, one can consider a modification of the above problem, in which $\mathcal{K}^n_\star$ is replaced by the class of axisymmetric bounded convex domains and in which the family of transformations $\mathsf{GL}_n$ is restricted to diagonal ones. In such a modified setting the conjecture is settled in Theorem 9 of [MR3551851] for $n = 2$.
• ### Discrete Faber-Krahn inequality

The classical Faber-Krahn inequality states that among all sets of equal volume the lowest Dirichlet eigenvalue $\lambda_1(\Omega)$ is minimized by the ball. The following longstanding conjecture concerns the corresponding problem about minimization of $\lambda_1(\Omega)$ among polygons.

#### Conjecture 2.3.

[D. Bucur] Among all polygons of a given area and having not more than $N$ edges the lowest Dirichlet eigenvalue $\lambda_1(\Omega)$ is minimized by the regular $N$-gon.
The conjecture is settled by G. Polya for $N=3, 4$ via Steiner symmetrization. However, if $N\geq 5$ symmetrization might increase the number of edges of a polygon and consequently this case requires a different approach. For all $N$, it is known that minimizers exist and have precisely $N$ edges \cite{}. \ADD

Interesting partial questions and further problems are for instance:
1. Prove that the regular $N$-gon is a local minimizer.
2. Prove that a minimizer must be convex.
3. Prove that $\lambda_1(\Omega_N)$ of the regular $N$-gon $\Omega_N$ is decreasing as a function of $N$. This is a necessary condition for the validity of the conjecture (remarked by C. Nitsch).
4. Consider the corresponding problem, but for the maximization of the first non-trivial Neumann eigenvalue.
5. Consider the corresponding problems but with the perimeter constraint.
For the Neumann problem, the case $N=3$ was solved by R. Laugesen and B. Siudeja [MR2567204] for both area and perimeter constraints, but for $N\geq 4$ the problem remains open.
1. Remark. [Richard Laugesen] For more on shape optimization for triangles, see Chapter 6 (pp. 149-200) of the book "Shape Optimization and Spectral Theory", edited by Antoine Henrot, 2017 (open access).
• ### van den Berg’s conjecture

Let $\Omega \subset \mathbb{R}^n$ be a convex domain and let $\rho, D$ denote its inradius and diameter, respectively. Let $u$ denote the eigenfunction corresponding to the first eigenvalue of the Dirichlet Laplacian on $\Omega$. According to a theorem due to G. Chiti [MR0652928] there exists a dimensional constant $C$ such that $$\|u\|_{L^\infty(\Omega)} \leq \frac{C}{\rho^{n/2}}\|u\|_{L^2(\Omega)}.$$ The following conjecture was posed by M. van den Berg [MR1804178]

#### Conjecture 2.4.

[S. Steinerberger] There is a dimensional constant $C$ such that $$\|u\|_{L^\infty(\Omega)} \leq \frac{C}{\rho^{n/2}}\Bigl(\frac{\rho}{D}\Bigr)^{1/6}\|u\|_{L^2(\Omega)}.$$
Thanks to work of B. Georgiev, M. Mukherjee, and S. Steinerberger the conjecture is known to be true when $n=2$ [MR3859539]. The exponent $1/6$ is what is expected from consideration of a cone. Proving the corresponding result with a different improvement in terms of $\rho/D$ would be interesting.
• ### Concavity of the principal Dirichlet eigenfunction

Let $\Omega\subset \mathbb{R}^n$ be a bouned convex domain and let $u$ be the principal eigenfunction of the Dirichlet Laplacian on $\Omega$. By a classical result of H. Brascamp and E. Lieb $u$ is log-concave [MR0450480]. Moreover, it is known that $u$ is power-concave meaning that $u^\alpha$ is concave in the usual sense for certain powers $\alpha > 0$. The best concavity exponent for a domain $\Omega$ is defined as $\alpha(\Omega) :=\sup\{\alpha \ge 0\colon u^\alpha \mbox{ is concave}\}.$ The following conjecture was formulated in [MR1280948] by P. Lindqvist.

#### Conjecture 2.5.

[A. Henrot] The best concavity exponent $\alpha(\Omega)$ is maximized by the ball.

Cite this as: AimPL: Shape optimization with surface interactions, available at http://aimpl.org/shapesurface.