The paper under review contains a proof of the following inequality. There is a universal constant \(c>0\) such that, if \(\Omega \subset \mathbb{R}^2\) is simply connected, \(u:\Omega \rightarrow \mathbb{R}\) vanishes on the boundary \(\partial \Omega\), and \(|u|\) assumes a maximum in \(x_0 \in \Omega\), then \[ \inf_{y \in \partial \Omega}{ \| x_0 - y\|} \geq c \left\| \frac{\Delta u}{u} \right\|^{-1/2}_{L^{\infty}(\Omega)}. \] The proof of this inequality (and higher dimensional results in the paper) is based on the probabilistic interpretation (Feynman-Kac formula). The authors give several applications of this inequality:
\(\bullet\) refined Hayman-Makai theorem: the point on the membrane that achieves the maximal amplitude is at distance \(\sim \lambda^{-1/2}\) from the boundary, where \(\lambda\) is the first Laplacian eigenfunction with Dirichlet boundary conditions;
\(\bullet\) generalization of the results proved by E. H. Lieb [Invent. Math. 74, 441–448 (1983; Zbl 0538.35058)], and B. Georgiev and M. Mukherjee [Anal. PDE, 11, 133–148 (2018; Zbl 1378.35208]): if \(u\) solves \(-\Delta u = Vu\) on \(\Omega \subset \mathbb{R}^n\) with Dirichlet boundary conditions, then the ball \(B\) with radius \(\sim \|V\|_{L^{\infty}(\Omega)}^{-1/2}\) centered at the point in which \(|u|\) assumes the maximum is almost fully contained in \(\Omega\) in the sense that \(|B \cap \Omega| \geq 0.99 |B|.\)
\(\bullet\) estimates of the torsion function;
\(\bullet\) a refinement of the Barta’s inequality [C.R. Acad. Sci. Paris 204, 472–473 (1937)].