Journal of Fourier Analysis and Applications

, Volume 15, Issue 6, pp 871–903 | Cite as

The Lp-Solvability of the Dirichlet Problem for Planar Elliptic Equations, Sharp Results

Article

Abstract

Assume that the elliptic operator L=div (A(x)) is Lp-resolutive, p>1, on the unit disc \(\mathbb{D}\subset \mathbb {R}^{2}\) . This means that the Dirichlet problem
$$\left\{\begin{array}{l@{\quad}l}Lu=0&\mbox{in }\mathbb{D},\\[3pt]u=g&\mbox{on }\partial\mathbb{D}\end{array}\right.$$
is uniquely solvable for any \(g\in L^{p}(\partial\mathbb{D})\) . Then, there exists ε>0 such that L is Lr- resolutive in the optimal range pε<r≤∞ (Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, Conference Board of the Mathematical Sciences, vol. 83, Am. Math. Soc., Providence, 1991). Here we determine the precise value of ε in terms of p and of a natural “norm” of the harmonic measure ωL.

Simultaneous solvability for couples of operators which are pull-back of the Laplacian under a quasiconformal mapping F and its inverse F−1 is also studied.

Finally we consider sequences of operators and study the weak convergence of their harmonic measures.

Keywords

Divergence elliptic equations Dirichlet problem Harmonic-measure Quasiconformal mappings 

Mathematics Subject Classification (2000)

35J25 35B65 35R05 

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References

  1. 1.
    Alberico, T.: Sharp estimates for the weighted maximal operator in Orlicz spaces. Rend. Accad. Sci. Fis. Mat. Napoli 74, 269–278 (2007) MathSciNetGoogle Scholar
  2. 2.
    Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton Univ. Press, Princeton (2008) Google Scholar
  3. 3.
    Beurling, A., Ahlfors, L.: The boundary correspondence under quasi-conformal mappings. Acta Math. 96, 125–142 (1956) CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Buckley, S.: Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Am. Math. Soc. 340(1), 253–272 (1993) CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Caffarelli, L., Fabes, E., Kenig, C.E.: Completely singular elliptic-harmonic measures. Indiana Univ. Math. J. 30(6), 917–924 (1981) CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Caffarelli, L., Fabes, E., Mortola, S., Salsa, S.: Boundary behavior of non-negative solutions of elliptic operators in divergence form. Indiana Univ. Math. J. 30(4), 621–640 (1981) CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Coifman, R.R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Stud. Math. 51, 241–250 (1974) MathSciNetMATHGoogle Scholar
  8. 8.
    Dahlberg, B.E.J.: On estimates of harmonic measure. Arch. Rat. Mech. Anal. 65, 272–288 (1977) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Dahlberg, B.E.J.: On the Poisson integral for Lipschitz and C 1 domains. Stud. Math. 66, 7–24 (1979) MathSciNetGoogle Scholar
  10. 10.
    Dahlberg, B.E.J.: On the absolute continuity of elliptic measures. Am. J. Math. 108(5), 1119–1138 (1986) CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    D’Apuzzo, L., Sbordone, C.: Reverse Hölder inequalities. A sharp result. Rend. Mat. Appl. 10(7), 357–366 (1990) MathSciNetMATHGoogle Scholar
  12. 12.
    De Giorgi, E., Spagnolo, S.: Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine. Boll. Unione Mat. Ital. 8(4), 391–411 (1973) MATHGoogle Scholar
  13. 13.
    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) CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Fefferman, R.: Large perturbations of elliptic operators and the solvability of the L p Dirichlet problem. J. Funct. Anal. 118, 477–510 (1993) CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Fefferman, R., Kenig, C.E., Pipher, J.: The theory of weights and the Dirichlet problem for elliptic equations. Ann. Math. 134, 65–124 (1991) CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Francfort, G.A., Murat, F.: Optimal bounds for conduction in two-dimensional, two phase, anisotropic media. In: Knops, R.J., Lacey, A.A. (eds.) Non-Classical Continuum Mechanics, Durham 1986. London Math. Soc. Lecture Note, vol. 122, pp. 197–212. Cambridge Univ. Press, Cambridge (1987) Google Scholar
  17. 17.
    Garcia-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland Math. Stud., vol. 116. North-Holland, Amsterdam (1985) MATHGoogle Scholar
  18. 18.
    Gehring, F.: The L p-integrability of the partial derivatives of a quasi-conformal mapping. Acta Math. 130, 265–277 (1973) CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Gotoh, Y.: On composition of operators which preserve BMO. Pac. J. Math. 201(2), 289–307 (2001) CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Iwaniec, T., Martin, G.: Geometric Function Theory and Nonlinear Analysis. Oxford Math. Monographs (2001) Google Scholar
  21. 21.
    Iwaniec, T., Sbordone, C.: Quasiharmonic fields. Ann. Inst. Poincaré Anal. Non-Linéaire 18(5), 519–572 (2001) CrossRefMathSciNetMATHGoogle Scholar
  22. 22.
    Johnson, R., Neugebauer, C.J.: Homeomorphisms preserving A p. Rev. Mat. Iberoam. 3(2), 249–273 (1987) MathSciNetMATHGoogle Scholar
  23. 23.
    Kenig, C.E.: Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems. Conference Board of the Mathematical Sciences, vol. 83. Am. Math. Soc., Providence (1991) Google Scholar
  24. 24.
    Kenig, C.E., Pipher, J.: The Neumann problem for elliptic equations with nonsmooth coefficients. Invent. Math. 113(3), 447–509 (1993) CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Kenig, C.E., Pipher, J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Math. 45(1), 199–217 (2001) MathSciNetMATHGoogle Scholar
  26. 26.
    Kenig, C.E., Koch, H., Pipher, J., Toro, T.: A new approach to absolute continuity of the elliptic measure, with applications to non-symmetric equations. Adv. Math. 153(2), 231–298 (2000) CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Korenovskii, A.A.: Mean Oscillations and Equimeasurable Rearrangements of Functions. Lecture Notes of Unione Matematica Italiana, vol. 4. Springer, Berlin (2007) MATHGoogle Scholar
  28. 28.
    Lehto, O.: Univalent Functions and Teichmüller Spaces. Graduate Texts in Mathematics, vol. 109. Springer, New York (1987) MATHGoogle Scholar
  29. 29.
    Lehto, O., Virtanen, K.: Quasiconformal Mappings in the Plane. Springer, Berlin (1971) Google Scholar
  30. 30.
    Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43(1), 126–166 (1938) CrossRefMathSciNetMATHGoogle Scholar
  31. 31.
    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972) CrossRefMathSciNetMATHGoogle Scholar
  32. 32.
    Murat, F., Tartar, L.: H-convergence. In: Topics in the Mathematical Modelling of Composite Materials. Progr. Nonlinear Differential Equations Appl., vol. 31, pp. 21–43. Birkhäuser, Boston (1997) Google Scholar
  33. 33.
    Radice, T.: New bounds for A weights. Ann. Acad. Sci. Fenn. Math. 33(1), 111–119 (2008) MathSciNetMATHGoogle Scholar
  34. 34.
    Sbordone, C.: Sharp embeddings for classes of weights and applications. Rend Accad. Naz. Sci. XL Mem. Mat. Appl. (5) 29(1), 339–354 (2005) MathSciNetGoogle Scholar
  35. 35.
    Spagnolo, S.: Some Convergence Problems, Convegno sulle Trasformazioni Quasiconformi e Questioni Connesse, INDAM, Rome, 1974. Symposia Mathematica, vol. 18, pp. 391–398. Academic Press, London (1976) Google Scholar
  36. 36.
    Vasyunin, V.: The sharp constant in the reverse Hölder inequality for Muckenhoupt weights. St. Petersb. Math. J. 15, 49–79 (2004) CrossRefMathSciNetGoogle Scholar
  37. 37.
    Zecca, G.: The unsolvability of the Dirichlet problem with L(log L)α boundary data. Rend. Accad. Sci. Fis. Mat. Napoli 72, 71–80 (2005) MathSciNetGoogle Scholar
  38. 38.
    Zecca, G.: On the Dirichlet problem with Orlicz boundary data. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10(3), 661–679 (2007) MathSciNetMATHGoogle Scholar

Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”Via Cintia—Complesso Universitario Monte S. AngeloNapoliItaly

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