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On the first eigenvalue of the generalized laplacian

Julian Fernandez Bonder, Ariel Salort

Abstract

In this work we investigate the energy of minimizers of Rayleigh-type quotients of the form $$ \frac{\int_ΩA(|\nabla u|)\, dx}{\int_ΩA(|u|)\, dx}. $$ These minimizers are eigenfunctions of the generalized laplacian defined as $Δ_a u = \text{div}\left(a(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right)$ where $a(t)=A'(t)$ and the Rayleigh quotient is comparable to the associated eigenvalue. On the function $A$ we only assume that it is a Young function but no $Δ_2$ condition is imposed. Since the problem is not homogeneous, the energy of minimizers is known to strongly depend on the normalization parameter $α=\int_ΩA(|u|)\, dx$. In this work we precisely analyze this dependence and show differentiability of the energy with respect to $α$ and, moreover, the limits as $α\to 0$ and $α\to \infty$ of the Rayleigh quotient. The nonlocal version of this problem is also analyzed.

On the first eigenvalue of the generalized laplacian

Abstract

In this work we investigate the energy of minimizers of Rayleigh-type quotients of the form These minimizers are eigenfunctions of the generalized laplacian defined as where and the Rayleigh quotient is comparable to the associated eigenvalue. On the function we only assume that it is a Young function but no condition is imposed. Since the problem is not homogeneous, the energy of minimizers is known to strongly depend on the normalization parameter . In this work we precisely analyze this dependence and show differentiability of the energy with respect to and, moreover, the limits as and of the Rayleigh quotient. The nonlocal version of this problem is also analyzed.
Paper Structure (14 sections, 23 theorems, 117 equations)

This paper contains 14 sections, 23 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Young functions
  4. Examples
  5. Orlicz and Orlicz-Sobolev spaces
  6. Orlicz spaces
  7. Orlicz-Sobolev spaces
  8. Fractional Orlicz-Sobolev spaces
  9. A review on the existence of minimizers for the energy function
  10. Elementary estimates on the energy function
  11. Continuity and differentiability of the energy function
  12. Sharp asymptotic behavior when $\alpha\to 0$ and $\alpha\to\infty$
  13. The behavior of $E(\alpha)/\alpha$ without the $\Delta_2-$condition
  14. Nonlocal case

Key Result

Proposition 2.1

Let $A$ be a Young function. Then, if $A\not\in \Delta_2^i$, $i=0,\infty$, we have that Moreover, if $A$ satisfies the slightly stronger assumption then $M_i(t) = 0$ if $0<t<1$.

Theorems & Definitions (43)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • ...and 33 more