2. The Dirichlet Laplacian

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 $\lambda_{k1}(\Omega_k^*)=\lambda_k(\Omega_k^*)$. 
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 \[ \sup\left\{ \inf_{T\in\mathsf{GL}_n} \lambda_1(T(K))\lambda_1((T(K))^\circ)\colon K\in\mathcal{K}^n_\star\right\}. \] 
Discrete FaberKrahn inequality
The classical FaberKrahn 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.
Interesting partial questions and further problems are for instance: Prove that the regular $N$gon is a local minimizer.
 Prove that a minimizer must be convex.
 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).
 Consider the corresponding problem, but for the maximization of the first nontrivial Neumann eigenvalue.
 Consider the corresponding problems but with the perimeter constraint.

Remark. [Richard Laugesen] For more on shape optimization for triangles, see Chapter 6 (pp. 149200) of the book "Shape Optimization and Spectral Theory", edited by Antoine Henrot, 2017 (open access).

van den Bergâ€™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 $$ \u\_{L^\infty(\Omega)} \leq \frac{C}{\rho^{n/2}}\Bigl(\frac{\rho}{D}\Bigr)^{1/6}\u\_{L^2(\Omega)}. $$ 
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 logconcave [MR0450480]. Moreover, it is known that $u$ is powerconcave 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.