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Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators

Authors: Steve Hofmann, Carlos Kenig, Svitlana Mayboroda and Jill Pipher
Journal: J. Amer. Math. Soc. 28 (2015), 483-529
MSC (2010): Primary 42B99, 42B25, 35J25, 42B20
Published electronically: May 21, 2014
MathSciNet review: 3300700
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Abstract: We consider divergence form elliptic operators $ L= {-}\textup {div}\, A(x) \nabla $, defined in the half space $ \mathbb{R}^{n+1}_+$, $ n\geq 2$, where the coefficient matrix $ A(x)$ is bounded, measurable, uniformly elliptic, $ t$-independent, and not necessarily symmetric. We establish square function/non-tangential maximal function estimates for solutions of the homogeneous equation $ Lu=0$, and we then combine these estimates with the method of ``$ \epsilon $-approximability'' to show that $ L$-harmonic measure is absolutely continuous with respect to surface measure (i.e., n-dimensional Lebesgue measure) on the boundary, in a scale-invariant sense: more precisely, that it belongs to the class $ A_\infty $ with respect to surface measure (equivalently, that the Dirichlet problem is solvable with data in $ L^p$, for some $ p<\infty $). Previously, these results had been known only in the case $ n=1$.

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  • [A] Pascal Auscher, On necessary and sufficient conditions for $ L^p$-estimates of Riesz transforms associated to elliptic operators on $ \mathbb{R}^n$ and related estimates, Mem. Amer. Math. Soc. 186 (2007), no. 871, xviii+75. MR 2292385 (2007k:42025),
  • [AA] Pascal Auscher and Andreas Axelsson, Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I, Invent. Math. 184 (2011), no. 1, 47-115. MR 2782252 (2012c:35111),
  • [AAAHK] M. Angeles Alfonseca, Pascal Auscher, Andreas Axelsson, Steve Hofmann, and Seick Kim, Analyticity of layer potentials and $ L^2$ solvability of boundary value problems for divergence form elliptic equations with complex $ L^\infty $ coefficients, Adv. Math. 226 (2011), no. 5, 4533-4606. MR 2770458 (2012g:35055),
  • [AHLMcT] Pascal Auscher, Steve Hofmann, Michael Lacey, Alan McIntosh, and Ph. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on $ {\mathbb{R}}^n$, Ann. of Math. (2) 156 (2002), no. 2, 633-654. MR 1933726 (2004c:47096c),
  • [AHLT] Pascal Auscher, Steve Hofmann, John L. Lewis, and Philippe Tchamitchian, Extrapolation of Carleson measures and the analyticity of Kato's square-root operators, Acta Math. 187 (2001), no. 2, 161-190. MR 1879847 (2004c:47096a),
  • [AT] Pascal Auscher and Philippe Tchamitchian, Square root problem for divergence operators and related topics, Astérisque 249 (1998), viii+172 (English, with English and French summaries). MR 1651262 (2000c:47092)
  • [BLP] Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam, 1978. MR 503330 (82h:35001)
  • [CFMS] L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30 (1981), no. 4, 621-640. MR 620271 (83c:35040),
  • [CF] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. MR 0358205 (50 #10670)
  • [CMS] R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304-335. MR 791851 (86i:46029),
  • [D] Björn E. J. Dahlberg, Approximation of harmonic functions, Ann. Inst. Fourier (Grenoble) 30 (1980), no. 2, vi, 97-107 (English, with French summary). MR 584274 (82i:31010)
  • [DJK] Björn E. J. Dahlberg, David S. Jerison, and Carlos E. Kenig, Area integral estimates for elliptic differential operators with nonsmooth coefficients, Ark. Mat. 22 (1984), no. 1, 97-108. MR 735881 (85h:35021),
  • [FS] C. Fefferman and E. M. Stein, $ H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137-193. MR 0447953 (56 #6263)
  • [G] John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1981. MR 628971 (83g:30037)
  • [GT] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190 (86c:35035)
  • [H] Steve Hofmann, A local $ Tb$ theorem for square functions, Perspectives in partial differential equations, harmonic analysis and applications, Proc. Sympos. Pure Math., vol. 79, Amer. Math. Soc., Providence, RI, 2008, pp. 175-185. MR 2500492 (2010b:42025)
  • [HKMP2] Hofmann S., Kenig C., Mayboroda S., and Pipher J., The Regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients. preprint.
  • [HLMc] Steve Hofmann, Michael Lacey, and Alan McIntosh, The solution of the Kato problem for divergence form elliptic operators with Gaussian heat kernel bounds, Ann. of Math. (2) 156 (2002), no. 2, 623-631. MR 1933725 (2004c:47096b),
  • [HMaMo] Hofmann S., Mayboroda S., and Mourgoglou M., $ L^p$ and endpoint solvability results for divergence form elliptic equations with complex $ L^{\infty }$ coefficients. preprint.
  • [HMiMo] Hofmann S., Mitrea M., and Morris A., The method of layer potentials in $ L^p$ and endpoint spaces for elliptic operators with $ L^\infty $ Coefficients. preprint.
  • [JK] David S. Jerison and Carlos E. Kenig, The Dirichlet problem in nonsmooth domains, Ann. of Math. (2) 113 (1981), no. 2, 367-382. MR 607897 (84j:35076),
  • [Ka] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473 (34 #3324)
  • [Ke] Carlos E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1994. MR 1282720 (96a:35040)
  • [KKPT] C. Kenig, H. Koch, J. Pipher, and T. Toro, A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, Adv. Math. 153 (2000), no. 2, 231-298. MR 1770930 (2002f:35071),
  • [KLS] Carlos E. Kenig, Fanghua Lin, and Zhongwei Shen, Homogenization of elliptic systems with Neumann boundary conditions, J. Amer. Math. Soc. 26 (2013), no. 4, 901-937. MR 3073881,
  • [KP] Carlos E. Kenig and Jill Pipher, The Neumann problem for elliptic equations with nonsmooth coefficients, Invent. Math. 113 (1993), no. 3, 447-509. MR 1231834 (95b:35046),
  • [KS1] Carlos E. Kenig and Zhongwei Shen, Homogenization of elliptic boundary value problems in Lipschitz domains, Math. Ann. 350 (2011), no. 4, 867-917. MR 2818717 (2012m:35039),
  • [KS2] Carlos E. Kenig and Zhongwei Shen, Layer potential methods for elliptic homogenization problems, Comm. Pure Appl. Math. 64 (2011), no. 1, 1-44. MR 2743875 (2011i:35009),
  • [R] Andreas Rosén, Layer potentials beyond singular integral operators, Publ. Mat. 57 (2013), no. 2, 429-454. MR 3114777,
  • [SW] James Serrin and H. F. Weinberger, Isolated singularities of solutions of linear elliptic equations, Amer. J. Math. 88 (1966), 258-272. MR 0201815 (34 #1697)
  • [V] Varopoulos N., A remark on BMO and bounded harmonic functions (1970).

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Additional Information

Steve Hofmann
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Carlos Kenig
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois, 60637

Svitlana Mayboroda
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455

Jill Pipher
Affiliation: Department of Mathematics, Brown University, Providence, Rhode Island 02912

Keywords: Divergence form elliptic equations, Dirichlet problem, harmonic measure, square function, non-tangential maximal function, $\epsilon$-approximability, $A_\infty$ Muckenhoupt weights, layer potentials.
Received by editor(s): February 10, 2012
Received by editor(s) in revised form: February 11, 2014
Published electronically: May 21, 2014
Additional Notes: Each of the authors was supported by the NSF
This work has been possible thanks to the support and hospitality of the University of Chicago, the University of Minnesota, the University of Missouri, Brown University, and the BIRS Centre in Banff (Canada). The authors would like to express their gratitude to these institutions.
Article copyright: © Copyright 2014 American Mathematical Society

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