The Dirichlet problem for the biharmonic equation in a Lipschitz domain

Dahlberg, Björn E. J. ; Kenig, C. E. ; Verchota, G. C.

Annales de l'Institut Fourier, Tome 36 (1986), p. 109-135 / Harvested from Numdam

Résumé

Dans cet article nous étudions le problème de Dirichlet pour l’opérateur biharmonique $\Delta ^2$, dans un domaine borné lipschitzien quelconque $D$ dans ${\bf R}^n$, et nous donnons des bornes optimales. Nous démontrons des résultats d’existence et d’unicité quand les valeurs au bord ont des dérivées dans $L^2(\partial D)$, et la dérivée normale appartient à $L^2(\partial D)$. La solution qu’on obtient prend les valeurs au bord dans le sens de la convergence non-tangentielle, et la fonction maximale non-tangentielle de $\nabla u$ appartient à $L^2(\partial D)$.

Abstract

In this paper we study and give optimal estimates for the Dirichlet problem for the biharmonic operator $\Delta ^2$, on an arbitrary bounded Lipschitz domain $D$ in ${\bf R}^n$. We establish existence and uniqueness results when the boundary values have first derivatives in $L^2(\partial D)$, and the normal derivative is in $L^2(\partial D)$. The resulting solution $u$ takes the boundary values in the sense of non-tangential convergence, and the non-tangential maximal function of $\nabla u$ is shown to be in $L^2(\partial D)$.

Détail

Publié le : 1986-01-01

MR 88a:35070 | Zbl 0589.35040 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1062

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