Shape optimization with surface interactions

    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 $\mathbb{R}^n$.

    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.

    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]

    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.