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On the uniqueness and simplicity of the principal eigenvalue

Marcello Lucia — 2005

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Given an open set Ω of R N N > 2 , bounded or unbounded, and a function w L N 2 Ω with w + 0 but allowed to change sign, we give a short proof that the positive principal eigenvalue of the problem - u = λ w x u , u D 0 1 , 2 Ω is unique and simple. We apply this result to study unbounded Palais-Smale sequences as well as to give a priori estimates on the set of critical points of functionals of the type I u = 1 2 Ω u 2 d x - Ω G x , u d x , u D 0 1 , 2 Ω , when G has a subquadratic growth at infinity.

One-dimensional symmetry of periodic minimizers for a mean field equation

Chang-Shou LinMarcello Lucia — 2007

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider on a two-dimensional flat torus T defined by a rectangular periodic cell the following equation Δ u + ρ e u T e u - 1 | T | = 0 , T u = 0 . It is well-known that the associated energy functional admits a minimizer for each ρ 8 π . The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting λ 1 ( T ) to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever ρ min { 8 π , λ 1 ( T ) | T | } . Our results hold more generally for solutions that are Steiner...

The p-Royden and p-Harmonic Boundaries for Metric Measure Spaces

Marcello LuciaMichael J. Puls — 2015

Analysis and Geometry in Metric Spaces

Let p be a real number greater than one and let X be a locally compact, noncompact metric measure space that satisfies certain conditions. The p-Royden and p-harmonic boundaries of X are constructed by using the p-Royden algebra of functions on X and a Dirichlet type problem is solved for the p-Royden boundary. We also characterize the metric measure spaces whose p-harmonic boundary is empty.

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