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Principal frequency of p-sub-Laplacians for general vector fields

Michael Ruzhansky, Bolys Sabitbek, Durvudkhan Suragan

Abstract

In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet p-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.

Principal frequency of p-sub-Laplacians for general vector fields

Abstract

In this paper, we prove the uniqueness and simplicity of the principal frequency (or the first eigenvalue) of the Dirichlet p-sub-Laplacian for general vector fields. As a byproduct, we establish the Caccioppoli inequalities and also discuss the particular cases on the Grushin plane and on the Heisenberg group.

Paper Structure

This paper contains 3 sections, 9 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Principal frequency of $p$-sub-Laplacians
  3. Caccioppoli inequalities for general vector fields

Key Result

Lemma 1.1

Let $\Omega \subset M$ be an open set. Let $p>1$. Let $u$ and $v$ be differentiable in a given set $\Omega$ with $v(x)\neq 0$ in $\Omega$. Define Then we have Moreover, we have $L(u,v)=0$ a.e. in $\Omega$ if and only if $u=cv$ a.e. in $\Omega$ with a positive constant $c$.

Theorems & Definitions (15)

  • Lemma 1.1
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lem_J']}
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['thm_eigenvalue']}
  • Corollary 2.3: Uniqueness
  • proof : Proof of Corollary \ref{['uniq']}
  • Theorem 2.4: Simplicity
  • proof : Proof of Theorem \ref{['simplicity']}
  • Theorem 2.5
  • ...and 5 more