On the characterization of the Dirichlet and Fucik spectra of the one-dimensional anisotropic p-Laplace operator

TL;DR

This paper characterizes the Dirichlet and Fucik spectra of a one-dimensional anisotropic p-Laplace operator, providing explicit descriptions and deriving a new, stronger Poincaré inequality, with novel results even for the linear case.

Contribution

It offers a complete characterization of the spectra for the anisotropic p-Laplace operator and introduces a new optimal Poincaré inequality, extending known results to anisotropic and nonlinear contexts.

Findings

Explicit characterization of the Dirichlet spectrum for a ≠ b

Derivation of a new, stronger Poincaré inequality

Complete description of the Fucik spectrum and solutions

Abstract

The paper is concerned with the Dirichlet spectrum $\Lambda^{a,b}_p(0,L)$ of the anisotropic $p$-Laplace operator $- \Delta^{a,b}_{p}$ on an interval $(0,L)$ where \[ \Delta^{a,b}_p u:= \left(a^{p}[(u')^{+}]^{p-1}-b^{p}[(u')^{-}]^{p-1}\right)', \ \ a, b > 0. \] The set $\Lambda^{a,b}_p(0,L)$ and the respective eigenfunctions are completely characterized for $a \neq b$ in terms of the corresponding ones within the isotropic context. As an interesting application, we derive a new optimal Poincar\'e inequality that is stronger than the classical counterpart. The leading ideas are based on glue arguments of conveniently modified eigenfunctions and maximum type principles. More generally, our approach allows to characterize the Fu\v c\'ik spectrum $\Sigma^{a,b}_p(0,L)$ of $- \Delta^{a,b}_{p}$ on $(0,L)$ and mainly the corresponding solutions. All results are novelty even for the nonlinear operator $\Delta^{a,b}_2$.