The Dirichlet problem without the maximum principle

Arendt, Wolfgang; ter Elst, A. F. M.

doi:10.5802/aif.3257

The Dirichlet problem without the maximum principle
[Le problème de Dirichlet sans le principe maximum]

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)

Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 763-782.

Résumé
Abstract

Considérons le problème de Dirichlet par rapport à un opérateur elliptique

A = - \sum_{k, l = 1}^{d} \partial_{k} a_{k l} \partial_{l} - \sum_{k = 1}^{d} \partial_{k} b_{k} + \sum_{k = 1}^{d} c_{k} \partial_{k} + c_{0}

sur un ensemble ouvert régulier de Wiener borné $Ω \subset ℝ^{d}$ , où $a_{k l}, c_{k} \in L_{\infty} (Ω, ℝ)$ et $b_{k}, c_{0} \in L_{\infty} (Ω, ℂ)$ . Supposons que $0$ n’est pas une valeur propre de $A$ avec conditions aux limites Dirichlet. Alors nous montrons que pour tout $φ \in C (\partial Ω)$ il existe un unique $u \in C (\bar{Ω}) \cap H_{loc}^{1} (Ω)$ tel que ${u |}_{\partial Ω} = φ$ et $A u = 0$ .

Dans le cas où $Ω$ a une frontière Lipschitz et $φ \in C (\bar{Ω}) \cap H^{1 / 2} (\bar{Ω})$ , nous montrons que $u$ coïncide avec la solution variationnelle dans $H^{1} (Ω)$ .

Consider the Dirichlet problem with respect to an elliptic operator

A = - \sum_{k, l = 1}^{d} \partial_{k} a_{k l} \partial_{l} - \sum_{k = 1}^{d} \partial_{k} b_{k} + \sum_{k = 1}^{d} c_{k} \partial_{k} + c_{0}

on a bounded Wiener regular open set $Ω \subset ℝ^{d}$ , where $a_{k l}, c_{k} \in L_{\infty} (Ω, ℝ)$ and $b_{k}, c_{0} \in L_{\infty} (Ω, ℂ)$ . Suppose that the associated operator on $L_{2} (Ω)$ with Dirichlet boundary conditions is invertible. Then we show that for all $φ \in C (\partial Ω)$ there exists a unique $u \in C (\bar{Ω}) \cap H_{loc}^{1} (Ω)$ such that ${u |}_{\partial Ω} = φ$ and $A u = 0$ .

In the case when $Ω$ has a Lipschitz boundary and $φ \in C (\bar{Ω}) \cap H^{1 / 2} (\bar{Ω})$ , then we show that $u$ coincides with the variational solution in $H^{1} (Ω)$ .

Reçu le : 2017-10-24
Révisé le : 2018-02-01
Accepté le : 2018-03-13
Publié le : 2019-06-03

Zbl

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

Affiliations des auteurs :

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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     title = {The {Dirichlet} problem without the maximum principle},
     journal = {Annales de l'Institut Fourier},
     pages = {763--782},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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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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