Area integral estimates for higher order elliptic equations and systems
Dahlberg, Björn E. J. ; Kenig, Carlos E. ; Pipher, Jill ; Verchota, G. C.
Annales de l'Institut Fourier, Tome 47 (1997), p. 1425-1461 / Harvested from Numdam

Soit L un système elliptique d’ordre m2 d’opérateurs différentiels homogènes. On établit l’équivalence entre la norme L p de la fonction maximale et la fonctionnelle quadratique des solutions de L dans les domaines lipschitziens. On donne quelques conséquences de ce résultat.

Let L be an elliptic system of higher order homogeneous partial differential operators. We establish in this article the equivalence in L p norm between the maximal function and the square function of solutions to L in Lipschitz domains. Several applications of this result are discussed.

@article{AIF_1997__47_5_1425_0,
     author = {Dahlberg, Bj\"orn E. J. and Kenig, Carlos E. and Pipher, Jill and Verchota, G. C.},
     title = {Area integral estimates for higher order elliptic equations and systems},
     journal = {Annales de l'Institut Fourier},
     volume = {47},
     year = {1997},
     pages = {1425-1461},
     doi = {10.5802/aif.1605},
     mrnumber = {98m:35045},
     zbl = {0892.35053},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1997__47_5_1425_0}
}
Dahlberg, Björn E. J.; Kenig, Carlos E.; Pipher, Jill; Verchota, G. C. Area integral estimates for higher order elliptic equations and systems. Annales de l'Institut Fourier, Tome 47 (1997) pp. 1425-1461. doi : 10.5802/aif.1605. http://gdmltest.u-ga.fr/item/AIF_1997__47_5_1425_0/

[1] V. Adolfsson and J. Pipher, The inhomogeneous Dirichlet problem for Δ2 in Lipschitz domains (preprint). | Zbl 0934.35038

[2] S. Agmon, Lectures on elliptic boundary value problems, D. Van Nostrand, Princeton, NJ, 1965. | MR 31 #2504 | Zbl 0142.37401

[3] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II., Comm. Pure Appl. Math., 17 (1964), 35-92. | MR 28 #5252 | Zbl 0123.28706

[4] R.M. Brown and Z. Shen, Boundary value problems in Lipschitz cylinders for three dimensional parabolic systems, Rivista Mat. Ibero., 8, no 3 (1992), 271-303. | MR 94b:35133 | Zbl 0782.35033

[5] R.M. Brown and Z. Shen, Estimates for the Stokes operator in Lipschitz domains, Ind. U. Math. J., 44, no 4 (1995), 1183-1206. | MR 97c:35152 | Zbl 0858.35098

[6] D. Burkholder and R. Gundy, Distribution function inequalities for the area integral, Studia Math., 44 (1972), 527-544. | MR 49 #5309 | Zbl 0219.31009

[7] M.D. Choi and T.Y. Lam, An old question of Hilbert, Queen's papers on pure and applied math 46 (1977), Queen's University Kingston, Ontario, 385-405. | MR 58 #16503 | Zbl 0382.12010

[8] R.R. Coifman, A. Mcintosh and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur L2 pour les courbes lipschitziennes, Ann. of Math., 116 (1982), 361-387. | MR 84m:42027 | Zbl 0497.42012

[9] B.E.J. Dahlberg, On the Poisson integral for Lipschitz and C1 domains, Studia Math., 66 (1979), 13-24. | MR 81g:31007 | Zbl 0422.31008

[10] B.E.J. 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 82f:31003 | Zbl 0449.31002

[11] B.E.J. Dahlberg, Poisson semigroups and singular integrals, Proc. A.M.S., 97, no 1 (1986), 41-48. | MR 87g:42035 | Zbl 0595.31009

[12] B.E.J. Dahlberg, D.S. Jerison and C.E. Kenig, Area integral estimates for elliptic differential operators with nonsmooth coefficients, Arkiv. Mat., 22 (1984), 97-108. | MR 85h:35021 | Zbl 0537.35025

[13] B.E.J. Dahlberg and C.E. Kenig, Lp estimates for the 3-dimensional systems of elastostatics on Lipschitz domains, Analysis and partial differential equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, (1990), 621-634. | MR 91h:35053 | Zbl 0702.35076

[14] B.E.J. Dahlberg, C.E. Kenig and G. Verchota, The Dirichlet problem for the biharmonic equation in a Lipschitz domain, Ann. Inst. Fourier, Grenoble, 36-3 (1986), 109-135. | Numdam | MR 88a:35070 | Zbl 0589.35040

[15] A. Douglis and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations, Comm. Pure Appl. Math., 8 (1955), 503-538. | MR 17,743b | Zbl 0066.08002

[16] C. Fefferman and E. Stein, HP spaces of several variables, Acta Math., 129 (1972), 137-193. | MR 56 #6263 | Zbl 0257.46078

[17] G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities, 2nd ed, Cambridge University Press, Cambridge, 1988. | MR 89d:26016 | Zbl 0010.10703

[18] S. Hofmann and J.L. Lewis, L2 solvability and representation by caloric layer potential in time-varying domains, Annals Math., 144 (1996), 349-420. | MR 97h:35072 | Zbl 0867.35037

[19] D.S. Jerison and C.E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219. | MR 96b:35042 | Zbl 0832.35034

[20] F. John, Plane waves and spherical means, Interscience Publishers, Inc., New York, 1955. | Zbl 0067.32101

[21] C.E. Kenig and E.M. Stein, Unpublished, communicated by C. E. Kenig.

[22] C. Li, A. Mcintosh and S. Semmes, Convolution singular integrals on Lipschitz surfaces, Journ. A.M.S., 5 (1992), 455-481. | MR 93b:42029 | Zbl 0763.42009

[23] T.S. Motzkin, The arithmetic - geometric inequality, Inequalities (O. Shisha, ed), Academic Press, New York, 1967, 205-224.

[24] J. Necas, Sur les domaines du type N, Czechoslovak. Math. J., 12 (1962), 274-287. | MR 27 #2709 | Zbl 0106.27001

[25] J. Pipher and G. Verchota, Area integral results for the biharmonic operator in Lipschitz domains, Trans. A. M. S., 327, no 2 (1991), 903-917. | MR 92a:35052 | Zbl 0774.35022

[26] J. Pipher and G. Verchota, The maximum principle for biharmonic functions in Lipschitz and C1 domains, Commentarii Math. Helvetici, 68 (1993), 385-414. | MR 94j:35030 | Zbl 0794.31005

[27] J. Pipher and G. Verchota, Dilation invariant estimates and the boundary Gårding inequality for higher order elliptic operators, Ann. of Math., 142 (1995), 1-38. | MR 96g:35052 | Zbl 0878.35035

[28] J. Pipher and G. Verchota, Maximum principles for the polyharmonic equation on Lipschitz domains, Potential Analysis, 4, no 6 (1995), 615-636. | MR 96i:35021 | Zbl 0844.35013

[29] Z. Shen, Resolvent estimates in Lp for elliptic systems in Lipschitz domains, J. Funct. Anal., 133, no 1 (1995), 224-251. | MR 96h:35045 | Zbl 0853.35015

[30] E.M. Stein, Singular Integrals and Differentiability Properties of functions, Princeton University Press, Princeton, N.J., 1970. | MR 44 #7280 | Zbl 0207.13501

[31] G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains, J. Funct. Anal., 59, no 3 (1984), 572-611. | MR 86e:35038 | Zbl 0589.31005

[32] G. Verchota, The Dirichlet problem for the polyharmonic equation in Lipschitz domains, Ind. Math. J., 39, no 3 (1990). | MR 91k:35073 | Zbl 0724.31005

[33] G. Verchota, Potential for the Dirichlet problem in Lipschitz domains, Potential Theory - ICPT94 (Král et al., eds.), Walter de Gruyter & Co., Berlin (1996), 167-187. | MR 97f:35041 | Zbl 0858.35033

[34] G.C. Verchota and A.L. Vogel, Nonsymmetric systems on nonsmooth planar domains, to appear, Trans. A.M.S.. | Zbl 0892.35055

[35] G.C. Verchota and A.L. Vogel, Nonsymmetric systems and area integral estimates, in preparation. | Zbl 0941.35022