The theory of De Giorgi for non-local operators

Moritz Kassmann

doi:10.1016/j.crma.2007.10.007

Partial Differential Equations

The theory of De Giorgi for non-local operators
[La théorie de De Giorgi pour les opérateurs non locaux]

Présenté par : Alain Bensoussan

Moritz Kassmann ¹

¹ Institut für Angewandte Mathematik, Beringstraße 6, D-53115 Bonn, Germany

Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 621-624.

Résumé
Abstract

Sous des conditions générales pour $k : R^{d} \times R^{d} \to [0, \infty)$ , nous étudions les opérateurs intégro-différentiels $L$ de type (1). Nos conditions pour k impliquent qu'il existe un $α \in (0, 2)$ tel que $k (x, y) {| x - y |}^{d + α}$ reste borné pour de petits $| x - y |$ . Soit $Ω \subset R^{d}$ un ouvert borné. Soit $D_{Ω} (E) = L^{\infty} (R^{d}) \cap H_{loc}^{α / 2} (Ω)$ . Une fonction $u \in D_{Ω} (E)$ est nommée $L$ -harmonique en Ω si pour tout $ϕ \in C_{0}^{\infty} (Ω)$ $E (u, ϕ) = 0$ . Le but de cette Note est de trouver des bornes locales pour des fonctions $L$ -harmoniques. Les principaux resultats démontrent que des fonctions $u \in D_{B} (E)$ qui sont $L$ -harmoniques dans la boule B satisfont des estimations a priori dans $C^{β} (\bar{B^{'}})$ pour un $β > 0$ et pour tout $B^{'} ⋐ B$ . Les résultats de ce travail peuvent être regardés comme une généralisation de la théorie dite De Giorgi–Nash–Moser aux opérateurs integro-differentiels d'ordre $α \in (0, 2)$ .

Under quite general assumptions on $k : R^{d} \times R^{d} \to [0, \infty)$ , we study integro-differential operators $L$ of the form

(L u) (x) = 2 \lim_{ε \to 0} \int_{\binom{y \in R^{d}}{| y - x | > ε}} (u (y) - u (x)) k (x, y) d y .

(1)

Our assumptions on k imply that there is

α \in (0, 2)

such that

k (x, y) {| x - y |}^{d + α}

stays bounded for small

| x - y |

. Let

Ω \subset R^{d}

be a bounded open set. Set

D_{Ω} (E) = L^{\infty} (R^{d}) \cap H_{loc}^{α / 2} (Ω)

. We call a function

u \in D_{Ω} (E)

L

-harmonic in Ω if for any

ϕ \in C_{0}^{\infty} (Ω)

E (u, ϕ) = \underset{R^{d} \times R^{d}}{\int \int} (u (y) - u (x)) (ϕ (y) - ϕ (x)) k (x, y) d x d y = 0 .

(2)

The aim of this Note is to prove local bounds for

L

-harmonic functions. The main results says that functions

u \in D_{B} (E)

which are

L

-harmonic in the ball B satisfy a priori estimates in

C^{β} (\bar{B^{'}})

for some

β > 0

and any

B^{'} ⋐ B

. The results can be seen as a generalization of the so-called De Giorgi–Nash–Moser theory to integro-differential operators of order

α \in (0, 2)

.

Reçu le : 2007-08-28
Accepté le : 2007-09-28
Publié le : 2007-11-19

DOI : 10.1016/j.crma.2007.10.007

Affiliations des auteurs :

Moritz Kassmann ¹

¹ Institut für Angewandte Mathematik, Beringstraße 6, D-53115 Bonn, Germany

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     title = {The theory of {De} {Giorgi} for non-local operators},
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Moritz Kassmann. The theory of De Giorgi for non-local operators. Comptes Rendus. Mathématique, Volume 345 (2007) no. 11, pp. 621-624. doi : 10.1016/j.crma.2007.10.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.007/

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