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Potential theory of non-divergence form elliptic equations

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1563)

Keywords

  • Maximum Principle
  • Elliptic Equation
  • Lipschitz Domain
  • Harmonic Measure
  • Harnack Inequality

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References

  1. A.D. Alexandrov, Uniqueness condition and estimates of the solution of Dirichlet's problem, Vestmik Leningrad 13 (1963), no. 3, 5–39.

    Google Scholar 

  2. I. Bakelman, Theory of quasilinear elliptic equations, Siberian Math J. 2 (1961), 179–186.

    MathSciNet  Google Scholar 

  3. R. Bass, The Dirichlet problem for radially homogeneous elliptic operators, Trans. AMS 320 (1990), 593–674.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. R. Bass and K. Burdzy, The boundary Harnack principle for non-divergence form elliptic operators, preprint.

    Google Scholar 

  5. B. Barcelo, L. Escauriaza and E. Fabes, Gradient estimates at the boundary for solutions to nondivergence elliptic equations, Contemp. Math. 107 (1990), 1–12.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. P. Bauman, Equivalence of the Green's functions for diffusion operators in ℝ n: a counterexample, Proc. AMS 91 (1984), 64–68.

    MathSciNet  MATH  Google Scholar 

  7. P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints, Ark. Mat. 22 (1984), 153–173.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. P. Bauman, A Wiener test for nondivergence structure second order elliptic equations, Indiana U. Math J. 4 (1985), 825–844.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. D. Burkholder and R. Gundy, Distribution function inequalities for the area integral, Studia Math. 44 (1972), 527–544.

    MathSciNet  MATH  Google Scholar 

  10. D. Burkholder, R. Gundy and M. Silverstein, A maximal function characterization of the class H p, Trans. AMS 157 (1971), 137–153.

    MathSciNet  MATH  Google Scholar 

  11. L. Caffarelli, Interior a priori estimates for solutions of fully non-linear equations, Annals of Math. 130 (1989), 189–213.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. L. Caffarelli, Interior W 2p estimates for solutions of Monge-Ampére equations, Annals of Math. 131 (1990), 135–150.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. L. Caffarelli, E. Fabes and C. Kenig, Completely singular elliptic-harmonic measures, Indiana U. Math. J. 30 (1981), 917–924.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. L. Caffarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana U. Math. J. 30 (1981), 621–640.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, Comm. Pure Appl. Math. 37 (1984), 369–402.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. A. Calderón Commutators of singular integral operators, Proc. Nat. Acad. Sci., USA 53 (1965), 1092–1099.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. C. Cerutti, Proc. AMS, to appear.

    Google Scholar 

  18. C. Cerutti, L. Escauriaza and E. Fabes, Uniqueness in the Dirichlet problem for some elliptic operators with discontinuous coefficients, Annali di Matematica Pura ed Aplicata, to appear.

    Google Scholar 

  19. C. Cerutti, L. Escauriaza and E. Fabes, Uniqueness for some diffusion with discontinuous coefficients, Annals of Probability, to appear.

    Google Scholar 

  20. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250.

    MathSciNet  MATH  Google Scholar 

  21. B. Dahlberg, Weighted norm inequalities for the Lusim area integral and non-tangential maximal function for functions harmonic in a Lipschitz domain, Studia Math. 67 (1980), 297–314.

    MathSciNet  MATH  Google Scholar 

  22. B. Dahlberg, D. Jerison and C. Kenig, Area integral estimates for elliptic differential operators with non-smooth coefficients, Ark. Mat. 22 (1984), 97–108.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. L. Escauriaza, Uniqueness in the Dirichlet problem for time independent elliptic operators, IMA Volumes in Math. and its Appl. 42 (1991), 115–127.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. L. Escauriaza and C. Kenig, Area integral estimates for solutions and normalized adjoint solutions to nondivergence form elliptic equations, Ark. Mat., to appear.

    Google Scholar 

  25. L.C. Evans, Classical solutions of fully nonlinear convex second order elliptic equations, Comm. Pure Appl. Math. 35 (1982), 333–363.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. L.C. Evans, Some estimates for nondivergence structure second order elliptic equations, Trans. AMS 287 (1985), 701–712.

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. E. Fabes, N. Garofalo, S. Marin-Malave and S. Salsa, Fatou theorems for some nonlinear elliptic equations, Revista Mat. Ibero-Americana 4 (1988), 227–251.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. E. Fabes and D. Stroock, The L p integrability of Green's functions and fundamental solutions for elliptic and parabolic equations, Duke Math. J. 51 (1984), 977–1016.

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. C. Fefferman and E. Stein, H p spaces of several variables, Acta. Math. 129 (1972), 137–193.

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. D. Gilbarg and J. Serrin, On isolated singularities of solutions of second order elliptic differential equations, J. D'Analyse Math. 4 (1955–1956), 309–340.

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. R. Gundy and R. Wheeden, Weighted integral inequalities for the non-tangential maximal function, Lusin area integral and Walsh-Paley series, Studia Math. 49 (1974), 107–124.

    MathSciNet  MATH  Google Scholar 

  32. D. Jerison and C. Kenig, Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. in Math. 46 (1982), 80–147.

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. N. Krylov, Some new results in the theory of nonlinear elliptic and parabolic equations, Proc. of the ICM, Berkeley, Ca, 1986, pp. 1101–1109.

    Google Scholar 

  34. N. Krylov and M. Safonov, An estimate of the probability that a diffusion process hits a set of positive measure, Soviet Math. Dokl. 20 (1979), 253–255.

    MATH  Google Scholar 

  35. N. Krylov and M. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR Izv. 16 (1981), 151–164.

    CrossRef  MATH  Google Scholar 

  36. E. Landis, s-capacity and the behavior of a solution of a second order elliptic equation with discontinuous coefficients in the neighborhood of a boundary point, Soviet Math. Dokl. 9 (1968), 582–586.

    MathSciNet  MATH  Google Scholar 

  37. F.L. Lin, Second derivative L p estimates for elliptic equations of nondivergent type, Proc. AMS 96 (1986), 447–451.

    MATH  Google Scholar 

  38. J. Manfredi and A. Weitsman, On the Fatou theorem for p-harmonic functions, Comm. in PDE 13 (1988), 651–688.

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. K. Miller, Nonequivalence of regular boundary points for the Laplace and nondivergence equations even with continuous coefficients, Ann. Scuola Norm. Sup. Pisa 24 (1970), 159–163.

    MathSciNet  MATH  Google Scholar 

  40. J. Moser, On Harnack's theorem for elliptic differential equations, Comm. in Pure Appl. Math. 13 (1960), 457–468.

    CrossRef  MATH  Google Scholar 

  41. C. Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. 74 (1966), 15–30.

    CrossRef  MathSciNet  MATH  Google Scholar 

  42. C. Pucci, Operatori ellittichi estremanti, Ann. Mat. Pura Appl. 72 (1966), 141–170.

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. C. Pucci and G. Talenti, Elliptic (second-order) partial differential equations with measurable coefficients and approximating integral equations, Adv. in Math. 19 (1976), 48–105.

    CrossRef  MathSciNet  Google Scholar 

  44. M. Safonov, Harnack's inequality for elliptic equations and the Hölder property of their solutions, J. Soviet Math. (1983), 851–863.

    Google Scholar 

  45. M. Safonov, Unimprovability of estimates of Hölder constant for solutions of linear elliptic equations with measurable coefficients, Math. USSR 60 (1988), 269–281.

    CrossRef  MathSciNet  MATH  Google Scholar 

  46. E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970.

    MATH  Google Scholar 

  47. E. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Math. Studies #63 (1970), Princeton Univ. Press, Princeton, NJ.

    CrossRef  MATH  Google Scholar 

  48. E. Stein, The development of square functions in the work of A. Zygmund, Conference on Harmonic Analysis in honor of A. Zygmund, Wadsworth International Group, International Mathematical Series (1981), 2–30.

    Google Scholar 

  49. D. Stroock and S. Varadhan, Multidimensional Diffusion Processes, Springer Verlag, Heidelberg, New York, 1979.

    MATH  Google Scholar 

  50. N. N. Uraltseva, Impossibility of W 2q bounds for multidimensional elliptic equations with discontinuous coefficients, Zap. Naveñ. Sem. Leningrad Ozdel Mat. Inst. Steklov (LOMi) 5 (1967). *** DIRECT SUPPORT *** A00I6B21 00008

    Google Scholar 

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Kenig, C.E. (1993). Potential theory of non-divergence form elliptic equations. In: Dell'Antonio, G., Mosco, U. (eds) Dirichlet Forms. Lecture Notes in Mathematics, vol 1563. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0074092

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  • DOI: https://doi.org/10.1007/BFb0074092

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