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) no. 3, pp. 109-135.

Dans cet article nous étudions le problème de Dirichlet pour l’opérateur biharmonique ${\Delta }^2$, dans un domaine borné lipschitzien quelconque $D$ dans ${\mathbf{R}}^n$, et nous donnons des bornes optimales. Nous démontrons des résultats d’existence et d’unicité quand les valeurs au bord ont des dérivées dans $L^2\left(\partial D\right)$, et la dérivée normale appartient à $L^2\left(\partial D\right)$. 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 $L^2\left(\partial D\right)$.

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 ${\mathbf{R}}^n$. We establish existence and uniqueness results when the boundary values have first derivatives in $L^2\left(\partial D\right)$, and the normal derivative is in $L^2\left(\partial D\right)$. 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 $L^2\left(\partial D\right)$.

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