The Dirichlet problem for the biharmonic equation in a Lipschitz domain
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 Δ 2 , dans un domaine borné lipschitzien quelconque D dans 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 (D), et la dérivée normale appartient à L 2 (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 u appartient à L 2 (D).

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

@article{AIF_1986__36_3_109_0,
     author = {Dahlberg, Bj\"orn E. J. and Kenig, C. E. and Verchota, G. C.},
     title = {The {Dirichlet} problem for the biharmonic equation in a {Lipschitz} domain},
     journal = {Annales de l'Institut Fourier},
     pages = {109--135},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {36},
     number = {3},
     year = {1986},
     doi = {10.5802/aif.1062},
     zbl = {0589.35040},
     mrnumber = {88a:35070},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1062/}
}
TY  - JOUR
AU  - Dahlberg, Björn E. J.
AU  - Kenig, C. E.
AU  - Verchota, G. C.
TI  - The Dirichlet problem for the biharmonic equation in a Lipschitz domain
JO  - Annales de l'Institut Fourier
PY  - 1986
SP  - 109
EP  - 135
VL  - 36
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1062/
DO  - 10.5802/aif.1062
LA  - en
ID  - AIF_1986__36_3_109_0
ER  - 
%0 Journal Article
%A Dahlberg, Björn E. J.
%A Kenig, C. E.
%A Verchota, G. C.
%T The Dirichlet problem for the biharmonic equation in a Lipschitz domain
%J Annales de l'Institut Fourier
%D 1986
%P 109-135
%V 36
%N 3
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1062/
%R 10.5802/aif.1062
%G en
%F AIF_1986__36_3_109_0
Dahlberg, Björn E. J.; Kenig, C. E.; Verchota, G. C. The Dirichlet problem for the biharmonic equation in a Lipschitz domain. Annales de l'Institut Fourier, Tome 36 (1986) no. 3, pp. 109-135. doi : 10.5802/aif.1062. https://aif.centre-mersenne.org/articles/10.5802/aif.1062/

[1] J. Cohen and J. Gosselin, The Dirichlet problem for the biharmonic equation in a C1 domain in the plane, Indiana U. Math. J., Vol 32, 5 (1983), 635-685. | MR | Zbl

[2] R. Coifman, A. Mc Intosh and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur L2 pour les courbes lipschitziennes, Annals of Math., 116 (1982), 361-387. | MR | Zbl

[3] R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-differentiels, Asterisque, 57 (1978). | MR | Zbl

[4] A. Cordoba and C. Fefferman, A weighted norm inequality for singular integrals, Studia Math., 57 (1976), 97-101. | MR | Zbl

[5] B. Dahlberg, On estimates of harmonic measure, Arch. for Rational Mech. and Anal., 65 (1977), 272-288. | MR | Zbl

[6] B. Dahlberg, On the Poisson integral for Lipschitz and C1 domains, Studia Math., 66 (1979), 13-24. | MR | Zbl

[7] B. Dahlberg, Weighted norm inequalities for the Lusin area integral and the non-tangential maximal functions for functions harmonic in a Lipschitz domain, Studia Math., 67 (1980), 297-314. | MR | Zbl

[8] B. Dahlberg and C. Kenig, Hardy spaces and the Lp Neumann problem for Laplace's equation in a Lipschitz domain, to appear in Annals of Math. | Zbl

[9] B. Dahlberg and C. Kenig, Area integral estimate for higher order boundary value problems on Lipschitz domains, in preparation.

[10] C. Fefferman and E. Stein, Hp space of several variables, Acta Math., 129 (1972), 137-193. | MR | Zbl

[11] D. Jerison and C. Kenig, The dirichlet problem in non-smooth domains, Annals of Math., 113 (1981), 367-382. | MR | Zbl

[12] D. Jerison and C. Kenig, The Neumann problem on Lipschitz domain, Bull AMS, Vol 4 (1981), 103-207. | MR | Zbl

[13] D. Jerison and C. Kenig, Boundary value problems on Lipschitz domain, MAA Studies in Mathematics, vol 23, Studies in Partial differential Equations, W. Littmann, editor (1982), 1-68. | MR | Zbl

[14] C. Kenig, Recent progress on boundary values problems on Lipschitz domain, Proc. of Symp. in Pure Math., Vol 43 (1985), 175-205. | MR | Zbl

[15] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 192 (1974), 261-274. | MR | Zbl

[16] J. Nečas, Les méthodes directes en théorie des équations elliptiques, Academia, Prague, 1967.

[17] E. Stein and G. Weiss, On the theory of harmonic functions of several variables, I, Acta Math., 103 (1960), 25-62. | MR | Zbl

[18] G.C. Verchota, Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains, Thesis, University of Minnesota (1982), J. of Functional Analysis, 59 (1984), 572-611. | Zbl

[19] G.C. Verchota, The Dirichlet problem for biharmonic functions in C1 domains, preprint. | Zbl

Cité par Sources :