Area integral estimates for higher order elliptic equations and systems
Annales de l'Institut Fourier, Volume 47 (1997) no. 5, pp. 1425-1461.

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.

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.

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     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},
     pages = {1425--1461},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
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     year = {1997},
     doi = {10.5802/aif.1605},
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     url = {http://archive.numdam.org/articles/10.5802/aif.1605/}
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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, Volume 47 (1997) no. 5, pp. 1425-1461. doi : 10.5802/aif.1605. http://archive.numdam.org/articles/10.5802/aif.1605/

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

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

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

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