Subelliptic $p$-Laplacian spectral problem for H\"ormander vector fields

TL;DR

This paper investigates the spectral properties of the subelliptic p-Laplacian associated with H"ormander vector fields, establishing key eigenvalue and eigenfunction characteristics using variational methods.

Contribution

It provides new results on the smallest eigenvalue, eigenfunction positivity, and regularity for the subelliptic p-Laplacian in the context of H"ormander vector fields.

Findings

Smallest eigenvalue derived and shown to be simple and isolated.

Proved positivity and H"older regularity of the first eigenfunction.

Determined the optimal constant for the L^p-Poincaré-Friedrichs inequality.

Abstract

Based on variational methods, we study the spectral problem for the subelliptic $p$-Laplacian arising from smooth H\"ormander vector fields. We derive the smallest eigenvalue, prove its simplicity and isolatedness, establish the positivity of the first eigenfunction and show H\"older regularity of eigenfunctions. Moreover, we determine the best constant for the $L^{p}$-Poincar\'e-Friedrichs inequality for H\"ormander vector fields as a byproduct.