The Dirichlet problem without the maximum principle
[Le problème de Dirichlet sans le principe maximum]
Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 763-782.

Considérons le problème de Dirichlet par rapport à un opérateur elliptique

A=- k,l=1 d k a kl l - k=1 d k b k + k=1 d c k k +c 0

sur un ensemble ouvert régulier de Wiener borné Ω d , où a kl ,c k L (Ω,) et b k ,c 0 L (Ω,). Supposons que 0 n’est pas une valeur propre de A avec conditions aux limites Dirichlet. Alors nous montrons que pour tout φC(Ω) il existe un unique uC(Ω ¯)H loc 1 (Ω) tel que u| Ω =φ et Au=0.

Dans le cas où Ω a une frontière Lipschitz et φC(Ω ¯)H 1/2 (Ω ¯), nous montrons que u coïncide avec la solution variationnelle dans H 1 (Ω).

Consider the Dirichlet problem with respect to an elliptic operator

A=- k,l=1 d k a kl l - k=1 d k b k + k=1 d c k k +c 0

on a bounded Wiener regular open set Ω d , where a kl ,c k L (Ω,) and b k ,c 0 L (Ω,). Suppose that the associated operator on L 2 (Ω) with Dirichlet boundary conditions is invertible. Then we show that for all φC(Ω) there exists a unique uC(Ω ¯)H loc 1 (Ω) such that u| Ω =φ and Au=0.

In the case when Ω has a Lipschitz boundary and φC(Ω ¯)H 1/2 (Ω ¯), then we show that u coincides with the variational solution in H 1 (Ω).

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DOI : 10.5802/aif.3257
Classification : 31C25, 35J05, 31B05
Keywords: Dirichlet problem, Wiener regular, holomorphic semigroup
Mot clés : Problème de Dirichlet, Wiener régulier, semigroupe holomorphe
Arendt, Wolfgang 1 ; ter Elst, A. F. M. 2

1 Institute of Applied Analysis University of Ulm Helmholtzstr. 18 89081 Ulm (Germany)
2 Department of Mathematics University of Auckland Private bag 92019 Auckland (New Zealand)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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Arendt, Wolfgang; ter Elst, A. F. M. The Dirichlet problem without the maximum principle. Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 763-782. doi : 10.5802/aif.3257. https://aif.centre-mersenne.org/articles/10.5802/aif.3257/

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