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Low eigenvalues of the $p-$Laplacian in general open sets

Lorenzo Brasco, Luca Briani, Francesca Prinari

Abstract

We consider the minmax Ljusternik-Schnirelmann levels of the constrained $p-$Dirichlet integral, on a general open set of the Euclidean space. We show that, whenever one of these levels lies below the threshold given by the $L^p$ Poincaré constant ``at infinity'', it actually defines an eigenvalue of the Dirichlet $p-$Laplacian. We also prove an exponential decay at infinity for the relevant eigenfunctions: this can be seen as a Šnol-Simon--type estimate for the nonlinear case. Finally, we exhibit some peculiar examples of unbounded open sets to which our main result applies.

Low eigenvalues of the $p-$Laplacian in general open sets

Abstract

We consider the minmax Ljusternik-Schnirelmann levels of the constrained Dirichlet integral, on a general open set of the Euclidean space. We show that, whenever one of these levels lies below the threshold given by the Poincaré constant ``at infinity'', it actually defines an eigenvalue of the Dirichlet Laplacian. We also prove an exponential decay at infinity for the relevant eigenfunctions: this can be seen as a Šnol-Simon--type estimate for the nonlinear case. Finally, we exhibit some peculiar examples of unbounded open sets to which our main result applies.
Paper Structure (18 sections, 15 theorems, 266 equations, 1 figure)

This paper contains 18 sections, 15 theorems, 266 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Beneath the essential spectrum
  3. Eigenvalues of the $p-$Laplacian
  4. Main result
  5. Plan of the paper
  6. Preliminaries
  7. Notation
  8. Some Poincaré inequalities
  9. A weighted Sobolev space
  10. Exponential fall-off of subsolutions
  11. Perturbed eigenvalue problems
  12. Proof of the Main Theorem
  13. Some examples
  14. Sets with a massive core
  15. Curved waveguides
  16. ...and 3 more sections

Key Result

Lemma 2.1

Let $1<p<\infty$ and let $E\subseteq\mathbb{R}^N$ be a measurable set with finite measure. Then, for every $\varphi\in C^\infty_0(\mathbb{R}^N)$ and every $\varepsilon>0$ we have: for a constant $C=C(N,p)>0$.

Figures (1)

  • Figure 1: The "infinite whip" of Lemma \ref{['lm:whip']}. This can be viewed as the tubular neighborhood of the curve in dashed line, which is "asymptotically straight" at infinity.

Theorems & Definitions (37)

  • Remark 1.1
  • Remark 1.2: Fall-off
  • Remark 1.3: Existence of eigenfunctions
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 27 more