Inventiones mathematicae

, Volume 184, Issue 1, pp 47–115

Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I

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Abstract

We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with L2 boundary data. The coefficients A may depend on all variables, but are assumed to be close to coefficients A0 that are independent of the coordinate transversal to the boundary, in the Carleson sense ‖AA0C defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour results under finiteness of ‖AA0C. Our methods yield full characterization of weak solutions, whose gradients have L2 estimates of a non-tangential maximal function or of the square function, via an integral representation acting on the conormal gradient, with a singular operator-valued kernel. Also, the non-tangential maximal function of a weak solution is controlled in L2 by the square function of its gradient. This estimate is new for systems in such generality, and even for real non-symmetric equations in dimension 3 or higher. The existence of a proof a priori to well-posedness, is also a new fact.

As corollaries, we obtain well-posedness of the Dirichlet, Neumann and Dirichlet regularity problems under smallness of ‖AA0C and well-posedness for A0, improving earlier results for real symmetric equations. Our methods build on an algebraic reduction to a first order system first made for coefficients A0 by the two authors and A. McIntosh in order to use functional calculus related to the Kato conjecture solution, and the main analytic tool for coefficients A is an operational calculus to prove weighted maximal regularity estimates.

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References

  1. 1.
    Albrecht, D.: Functional calculi of commuting unbounded operators. Ph.D. Thesis, Monash University (1994) Google Scholar
  2. 2.
    Albrecht, D., Duong, X., McIntosh, A.: Operator theory and harmonic analysis. In: Instructional Workshop on Analysis and Geometry, Part III, Canberra, 1995. Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 34, pp. 77–136. Austral. Nat. Univ., Canberra (1996) Google Scholar
  3. 3.
    Albrecht, D., Franks, E., McIntosh, A.: Holomorphic functional calculi and sums of commuting operators. Bull. Aust. Math. Soc. 58(2), 291–305 (1998) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Alfonseca, M., Auscher, P., Axelsson, A., Hofmann, S., Kim, S.: Analyticity of layer potentials and L 2 solvability of boundary value problems for divergence form elliptic equations with complex L coefficients. Preprint at arXiv:0705.0836v1 [math.AP]
  5. 5.
    Auscher, P.: On Hofmann’s bilinear estimate. Math. Res. Lett. 16(5), 753–760 (2009) MATHMathSciNetGoogle Scholar
  6. 6.
    Auscher, P., Axelsson, A.: Weighted maximal regularity estimates and solvability of non-smooth elliptic systems II. Preprint Google Scholar
  7. 7.
    Auscher, P., Axelsson, A., Hofmann, S.: Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems. J. Funct. Anal. 255(2), 374–448 (2008) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Auscher, P., Axelsson, A., McIntosh, A.: On a quadratic estimate related to the Kato conjecture and boundary value problems. Contemp. Math. 205, 105–129 (2010) MathSciNetGoogle Scholar
  9. 9.
    Auscher, P., Axelsson, A., McIntosh, A.: Solvability of elliptic systems with square integrable boundary data. Ark. Mat. 48, 253–287 (2010) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Auscher, P., Hofmann, S., Lacey, M., McIntosh, A., Tchamitchian, P.: The solution of the Kato square root problem for second order elliptic operators on R n. Ann. Math. (2) 156(2), 633–654 (2002) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Auscher, P., Tchamitchian, P.: Calcul fontionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux). Ann. Inst. Fourier (Grenoble) 45(3), 721–778 (1995) MATHMathSciNetGoogle Scholar
  12. 12.
    Axelsson, A.: Non unique solutions to boundary value problems for non symmetric divergence form equations. Trans. Am. Math. Soc. 362, 661–672 (2010) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Axelsson, A., Keith, S., McIntosh, A.: Quadratic estimates and functional calculi of perturbed Dirac operators. Invent. Math. 163(3), 455–497 (2006) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Barton, A.: Elliptic partial differential equations with complex coefficients. Ph.D. Thesis, University of Chicago (2010). Preprint at arXiv:0911.2513v1 [math.AP]
  15. 15.
    Caffarelli, L., Fabes, E., Kenig, C.E.: Completely singular elliptic-harmonic measures. Indiana Univ. Math. J. 30(6), 917–924 (1981) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Coifman, R.R., McIntosh, A., Meyer, Y.: L’intégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes lipschitziennes. Ann. Math. (2) 116(2), 361–387 (1982) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Coifman, R.R., Meyer, Y., Stein, E.M.: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62(2), 304–335 (1985) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Dahlberg, B.: Estimates of harmonic measure. Arch. Ration. Mech. Anal. 65(3), 275–288 (1977) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Dahlberg, B.: On the absolute continuity of elliptic measures. Am. J. Math. 108(5), 1119–1138 (1986) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Dahlberg, B., Jerison, D., Kenig, C.: Area integral estimates for elliptic differential operators with nonsmooth coefficients. Ark. Mat. 22(1), 97–108 (1984) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Dahlberg, B., Kenig, C., Pipher, J., Verchota, G.: Area integral estimates for higher order elliptic equations and systems. Ann. Inst. Fourier (Grenoble) 47(5), 1425–1461 (1997) MATHMathSciNetGoogle Scholar
  22. 22.
    de Simon, L.: Un’applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine. Rend. Semin. Mat. Univ. Padova 34, 205–223 (1964) MATHGoogle Scholar
  23. 23.
    Dindos, M., Kenig, C., Pipher, J.: BMO solvability and the A condition for elliptic operators. Preprint at arXiv:1007.5496v1 [math.AP]
  24. 24.
    Dindos, M., Petermichl, S., Pipher, J.: The L p Dirichlet problem for second order elliptic operators and a p-adapted square function. J. Funct. Anal. 249(2), 372–392 (2007) CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Dindos, M., Rule, D.: Elliptic equations in the plane satisfying a Carleson condition. Preprint Google Scholar
  26. 26.
    Fabes, E., Jerison, D., Kenig, C.: Necessary and sufficient conditions for absolute continuity of elliptic-harmonic measure. Ann. Math. (2) 119(1), 121–141 (1984) CrossRefMathSciNetGoogle Scholar
  27. 27.
    Fefferman, R.: Large perturbations of elliptic operators and the solvability of the L p Dirichlet problem. J. Funct. Anal. 118(2), 477–510 (1993) CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Fefferman, R.A., Kenig, C.E., Pipher, J.: The theory of weights and the Dirichlet problem for elliptic equations. Ann. Math. 134(1), 65–124 (1991) CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Jerison, D.S., Kenig, C.E.: The Dirichlet problem in nonsmooth domains. Ann. Math. (2) 113(2), 367–382 (1981) CrossRefMathSciNetGoogle Scholar
  30. 30.
    Kenig, C.: Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. CBMS Regional Conference Series in Mathematics, vol. 83. American Mathematical Society, Providence (1994) MATHGoogle Scholar
  31. 31.
    Kenig, C., Koch, H., Pipher, J., Toro, T.: A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations. Adv. Math. 153(2), 231–298 (2000) CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Kenig, C., Meyer, Y.: Kato’s square roots of accretive operators and Cauchy kernels on Lipschitz curves are the same. In: Recent Progress in Fourier Analysis, El Escorial, 1983. North-Holland Math. Stud., vol. 111, pp. 123–143. North-Holland, Amsterdam (1985) CrossRefGoogle Scholar
  33. 33.
    Kenig, C., Pipher, J.: The Neumann problem for elliptic equations with nonsmooth coefficients. Invent. Math. 113(3), 447–509 (1993) CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Kenig, C., Pipher, J.: The Neumann problem for elliptic equations with nonsmooth coefficients. II. Duke Math. J. 81(1), 227–250 (1995) CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Kenig, C., Pipher, J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Mat. 45(1), 199–217 (2001) MATHMathSciNetGoogle Scholar
  36. 36.
    Kenig, C., Rule, D.: The regularity and Neumann problem for non-symmetric elliptic operators. Trans. Am. Math. Soc. 361(1), 125–160 (2009) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Kilty, J., Shen, Z.: The L p regularity problem on Lipschitz domains. Trans. Am. Math. Soc. (to appear) Google Scholar
  38. 38.
    Kunstmann, P., Weis, L.: Maximal L p-regularity for parabolic equations, Fourier multiplier theorems and H -functional calculus. In: Functional Analytic Methods for Evolution Equations. Lecture Notes in Math., vol. 1855, pp. 65–311. Springer, Berlin (2004) Google Scholar
  39. 39.
    Lancien, F., Lancien, G., Le Merdy, C.: A joint functional calculus for sectorial operators with commuting resolvents. Proc. Lond. Math. Soc. (3) 77(2), 387–414 (1998) CrossRefGoogle Scholar
  40. 40.
    Lancien, G., Le Merdy, C.: A generalized H functional calculus for operators on subspaces of L p and application to maximal regularity. Ill. J. Math. 42(3), 470–480 (1998) MATHGoogle Scholar
  41. 41.
    Lim, N.: The L p Dirichlet problem for divergence form elliptic operators with non-smooth coefficients. J. Funct. Anal. 138(2), 502–543 (1996) CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Rios, C.: L p regularity of the Dirichlet problem for elliptic equations with singular drift. Publ. Mat. 50(2), 475–507 (2006) MATHMathSciNetGoogle Scholar
  43. 43.
    Shen, Z.: A relationship between the Dirichlet and regularity problems for elliptic equations. Math. Res. Lett. 14(2), 205–213 (2007) MATHMathSciNetGoogle Scholar
  44. 44.
    Šneĭberg, I.J.: Spectral properties of linear operators in interpolation families of Banach spaces. Mat. Issled. 2(32), 214–229, 254–255 (1974) Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Univ. Paris-Sud, laboratoire de MathématiquesUMR 8628OrsayFrance
  2. 2.CNRSOrsayFrance
  3. 3.Centre for Mathematics and Its ApplicationsAustralian National UniversityCanberraAustralia
  4. 4.Matematiska InstitutionenLinköpings UniversitetLinköpingSweden

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