## Elliptic equations in the plane satisfying a Carleson measure condition.(English)Zbl 1206.35101

Summary: We settle (in dimension $$n=2$$) the open question whether for a divergence form equation div$$(A\nabla u) = 0$$ with coefficients satisfying certain minimal smoothness assumption (a Carleson measure condition), the $$L^p$$ Neumann and Dirichlet regularity problems are solvable for some values of $$p\in (1,\infty)$$. The related question for the $$L^p$$ Dirichlet problem was settled (in any dimension) in 2001 by C. E. Kenig and J. Pipher [Publ. Mat., Barc. 45, No. 1, 199–217 (2001; Zbl 1113.35314)].

### MSC:

 35J60 Nonlinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 35J67 Boundary values of solutions to elliptic equations and elliptic systems 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 35B65 Smoothness and regularity of solutions to PDEs

Zbl 1113.35314
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### References:

 [1] Coifman, R.R., Meyer, Y. and Stein, E.M.: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62 (1985), no. 2, 304-335. · Zbl 0569.42016 [2] Dahlberg, B.E.J.: On the Poisson integral for Lipschitz and $$C^1$$-domains. Studia Math. 66 (1979), no. 1, 13-24. · Zbl 0422.31008 [3] Dahlberg, B.E.J.: Approximation of harmonic functions. Ann. Inst. Fourier (Grenoble) 30 (1980), no. 2, 97-107. · Zbl 0417.31005 [4] Dahlberg, B.E.J.: Weighted norm inequalities for the Lusin area integral and the nontangential maximal functions for functions harmonic in a Lipschitz domain. Sudia Math. 67 (1980), no. 3, 297-314. · Zbl 0449.31002 [5] Dahlberg, B.E.J., Kenig, C.E., Pipher, J. and Verchota, G.C.: Area integral estimates for higher order elliptic equations and systems. Ann. Inst. Fourier (Grenoble) 47 (1997), no. 5, 1425-1461. · Zbl 0892.35053 [6] Dindoš, M., Petermichl, S. and Pipher, J.: The $$L^p$$ Dirichlet problem for second order elliptic operators and a $$p$$-adapted square function. J. Funct. Anal. 249 (2007), no. 2, 372-392. · Zbl 1174.35025 [7] Gilbarg, D. and Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order . Springer-Verlag, Heidelberg, 1997. · Zbl 1042.35002 [8] Grafakos, L.: Classical and Modern Fourier Analysis . Pearson Education, Upper Saddle River, NJ, 2004. · Zbl 1148.42001 [9] Jerison, D.S. and Kenig, C.E.: The Neumann problem on Lipschitz domains. Bull. Amer. Math. Soc. 4 (1981), no. 2, 203-207. · Zbl 0471.35026 [10] Kenig, C.E., Koch, H., Pipher, J. and Toro, T.: A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations. Adv. Math. 153 (2000), no. 2, 231-298. · Zbl 0958.35025 [11] Kenig, C.E. and Pipher, J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Mat. 45 (2001), no. 1, 199-217. · Zbl 1113.35314 [12] Kenig, C.E. and Rule, D.J.: The regularity and Neumann problem for non-symmetric elliptic operators. Trans. Amer. Math. Soc. 361 (2009), no. 1, 125-160. · Zbl 1178.35155 [13] Nečas, J.: Les méthodes directes en théorie des équations elliptiques . Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967. · Zbl 1225.35003 [14] Pipher, J.: Littlewood-Paley estimates: Some applications to elliptic boundary value problems. In Partial differential equations and their applications (Toronto, ON, 1995) , 221-238. CRM Proc. Lecture Notes 12 . Amer. Math. Soc., Providence, RI, 1997. · Zbl 0945.35028 [15] Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals . Princeton Mathematical Series 43 . Princeton University Press, Princeton, NJ, 1993. · Zbl 0821.42001
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