Harnack inequalities for quasi-minima of variational integrals

Abstract

L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/locate/anihpc) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Harnack inequalities for quasi-minima of variational integrals E. DI BENEDETTO

Remarks on the Harnak Inequality for Local-Minima of Scalar Integral Functionals with General Growth Conditions

“…In [26], G. Moscariello and L. Nania has obtain a results of hölder continuity for the local-minima of functional of the type (1.1) under the hypothesis that (1.4) holds with 1 < p ≤ m < ((Np)/(N-p)). In [17], G. M. Lieberman proved an Harnack inequality for the local-∈ minima of the functional (1.1) with Φ C² suth that verifies the following relation with 0 < c 5 < c 6 . We are interested in functionals with quasi-linear growths and we will proof a regularity result which extend the ones obtained in [17,24,26] to a wider N-functional class.…”

“…The proof of the Harnack inequality uses the techniques introduced in [6,17] and [24]. The only present novelty in the demonstrative technique is the use of an ɛ-Young inequality.…”

Power-type quasiminimizers

“…Tolksdorf [22] obtained a Caccioppoli inequality and a convexity result for quasiminimizers. The results in [10], [11], [12] and [25] were extended to metric spaces by Kinnunen-Shanmugalingam [16] and J. Björn [8] in the beginning of this century, see also A. Björn-Marola [6]. Soon afterwards, Kinnunen-Martio [15] showed that quasiminimizers have an interesting potential theory, in particular they introduced quasisuperharmonic functions, which are related to quasisuperminimizers in a similar way as superharmonic functions are related to supersolutions, see Definition 2.1.…”

“…They realized that De Giorgi's method could be extended to quasiminimizers, obtaining, in particular, local Hölder continuity. DiBenedetto and Trudinger [10] proved the Harnack inequality for quasiminimizers, as well as weak Harnack inequalities for quasisub-and quasisuperminimizers. We recall that a function u ∈ W 1,p loc (Ω) is a quasisub(super )minimizer if (1.1) holds for all nonpositive (nonnegative) ϕ ∈ W 1,p 0 (Ω).…”

“…We recall that a function u ∈ W 1,p loc (Ω) is a quasisub(super )minimizer if (1.1) holds for all nonpositive (nonnegative) ϕ ∈ W 1,p 0 (Ω). After the papers by Giaquinta-Giusti [11], [12] and DiBenedetto-Trudinger [10], Ziemer [25] gave a Wiener-type criterion sufficient for boundary regularity for quasiminimizers. Tolksdorf [22] obtained a Caccioppoli inequality and a convexity result for quasiminimizers.…”