Harnack inequalities for quasi-minima of variational integrals

Abstract

In his fundamental work on linear elliptic equations, De Giorgi established local bounds and Hölder estimates for functions satisfying certain integral inequalities. The main result of this paper is that the Harnack inequality can be proved directly for functions in the De Giorgi classes. This implies that every non-negative Q-minimum (in the terminology of Giaquinta and Giusti) satisfies a Harnack inequality. Résumé: Dans son travail fondamental sur les équations linéaires elliptiques, De Giorgi a donné des estimations locales et hölderiennes pour des fonctions satisfaisant certaines inégalités intégrales. Le résultat principal de cet article est que l’inégalité de Harnack peut être démontrée directement pour les fonctions appartenant aux classes de De Giorgi. Ceci implique que tout Q-minimum (au sens de Giaquinta et Giusti) non-négatif vérifie une inégalité de Harnack.

Keywords

Funding Information

National Science Foundation (48–206-80, MCS 8300293)

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