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Power-logconcavity of the Laplacian ground state

Graziano Crasta, Ilaria Fragalà

Abstract

Let $u$ be the first Dirichlet Laplacian eigenfunction of a bounded convex set $Ω$ in $\mathbb{R}^n$. We strengthen the classical result by Brascamp-Lieb which asserts that $u$ is logconcave in $Ω$: we prove that, if $u$ is normalized so that its $L^\infty$-norm does not exceed a threshold $\overlineκ (Ω)<1$ depending explicitly on the diameter of the domain and on its principal frequency, the function $- ( - \log u ) ^{1/2}$ is concave in $Ω$.

Power-logconcavity of the Laplacian ground state

Abstract

Let be the first Dirichlet Laplacian eigenfunction of a bounded convex set in . We strengthen the classical result by Brascamp-Lieb which asserts that is logconcave in : we prove that, if is normalized so that its -norm does not exceed a threshold depending explicitly on the diameter of the domain and on its principal frequency, the function is concave in .
Paper Structure (4 sections, 8 theorems, 82 equations, 1 figure)

This paper contains 4 sections, 8 theorems, 82 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['t:12log']}
  4. Proof of Theorem \ref{['t:level']}

Key Result

Theorem 1

Let $\Omega$ be an open bounded convex subset of $\mathbb{R} ^n$, and let $u$ be its first Dirichlet Laplacian eigenfunction, normalized so that $\max _{\overline \Omega } u = 1$. Then $u$ is $(1/2)$-logconcave. More precisely, if $\lambda _ 1 (\Omega)$ and $D _\Omega$ denote respectively the first for every $\kappa \in (0, \overline \kappa (\Omega)]$ it holds that

Figures (1)

  • Figure 1: Plot of the maps $s \mapsto \Psi _\kappa ( s)$ (from the right to the left, for $\kappa= \frac{1}{2}, {\frac{1}{ \sqrt 2}}, {\frac{ \sqrt 2}{ \sqrt 3}}, 1$), and in dashed line of the constant map $s \mapsto \frac{\pi ^ 2}{\lambda _ 1 (\Omega) D _\Omega ^ 2 }$.

Theorems & Definitions (13)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Proposition 7
  • proof
  • ...and 3 more