Keywords
- Maximum Principle
- Elliptic Equation
- Lipschitz Domain
- Harmonic Measure
- Harnack Inequality
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References
A.D. Alexandrov, Uniqueness condition and estimates of the solution of Dirichlet's problem, Vestmik Leningrad 13 (1963), no. 3, 5–39.
I. Bakelman, Theory of quasilinear elliptic equations, Siberian Math J. 2 (1961), 179–186.
R. Bass, The Dirichlet problem for radially homogeneous elliptic operators, Trans. AMS 320 (1990), 593–674.
R. Bass and K. Burdzy, The boundary Harnack principle for non-divergence form elliptic operators, preprint.
B. Barcelo, L. Escauriaza and E. Fabes, Gradient estimates at the boundary for solutions to nondivergence elliptic equations, Contemp. Math. 107 (1990), 1–12.
P. Bauman, Equivalence of the Green's functions for diffusion operators in ℝ n: a counterexample, Proc. AMS 91 (1984), 64–68.
P. Bauman, Positive solutions of elliptic equations in nondivergence form and their adjoints, Ark. Mat. 22 (1984), 153–173.
P. Bauman, A Wiener test for nondivergence structure second order elliptic equations, Indiana U. Math J. 4 (1985), 825–844.
D. Burkholder and R. Gundy, Distribution function inequalities for the area integral, Studia Math. 44 (1972), 527–544.
D. Burkholder, R. Gundy and M. Silverstein, A maximal function characterization of the class H p, Trans. AMS 157 (1971), 137–153.
L. Caffarelli, Interior a priori estimates for solutions of fully non-linear equations, Annals of Math. 130 (1989), 189–213.
L. Caffarelli, Interior W 2p estimates for solutions of Monge-Ampére equations, Annals of Math. 131 (1990), 135–150.
L. Caffarelli, E. Fabes and C. Kenig, Completely singular elliptic-harmonic measures, Indiana U. Math. J. 30 (1981), 917–924.
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.
L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations, Comm. Pure Appl. Math. 37 (1984), 369–402.
A. Calderón Commutators of singular integral operators, Proc. Nat. Acad. Sci., USA 53 (1965), 1092–1099.
C. Cerutti, Proc. AMS, to appear.
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.
C. Cerutti, L. Escauriaza and E. Fabes, Uniqueness for some diffusion with discontinuous coefficients, Annals of Probability, to appear.
R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250.
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.
B. Dahlberg, D. Jerison and C. Kenig, Area integral estimates for elliptic differential operators with non-smooth coefficients, Ark. Mat. 22 (1984), 97–108.
L. Escauriaza, Uniqueness in the Dirichlet problem for time independent elliptic operators, IMA Volumes in Math. and its Appl. 42 (1991), 115–127.
L. Escauriaza and C. Kenig, Area integral estimates for solutions and normalized adjoint solutions to nondivergence form elliptic equations, Ark. Mat., to appear.
L.C. Evans, Classical solutions of fully nonlinear convex second order elliptic equations, Comm. Pure Appl. Math. 35 (1982), 333–363.
L.C. Evans, Some estimates for nondivergence structure second order elliptic equations, Trans. AMS 287 (1985), 701–712.
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.
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.
C. Fefferman and E. Stein, H p spaces of several variables, Acta. Math. 129 (1972), 137–193.
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.
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.
D. Jerison and C. Kenig, Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. in Math. 46 (1982), 80–147.
N. Krylov, Some new results in the theory of nonlinear elliptic and parabolic equations, Proc. of the ICM, Berkeley, Ca, 1986, pp. 1101–1109.
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.
N. Krylov and M. Safonov, A certain property of solutions of parabolic equations with measurable coefficients, Math. USSR Izv. 16 (1981), 151–164.
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.
F.L. Lin, Second derivative L p estimates for elliptic equations of nondivergent type, Proc. AMS 96 (1986), 447–451.
J. Manfredi and A. Weitsman, On the Fatou theorem for p-harmonic functions, Comm. in PDE 13 (1988), 651–688.
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.
J. Moser, On Harnack's theorem for elliptic differential equations, Comm. in Pure Appl. Math. 13 (1960), 457–468.
C. Pucci, Limitazioni per soluzioni di equazioni ellittiche, Ann. Mat. Pura Appl. 74 (1966), 15–30.
C. Pucci, Operatori ellittichi estremanti, Ann. Mat. Pura Appl. 72 (1966), 141–170.
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.
M. Safonov, Harnack's inequality for elliptic equations and the Hölder property of their solutions, J. Soviet Math. (1983), 851–863.
M. Safonov, Unimprovability of estimates of Hölder constant for solutions of linear elliptic equations with measurable coefficients, Math. USSR 60 (1988), 269–281.
E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, NJ, 1970.
E. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Annals of Math. Studies #63 (1970), Princeton Univ. Press, Princeton, NJ.
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.
D. Stroock and S. Varadhan, Multidimensional Diffusion Processes, Springer Verlag, Heidelberg, New York, 1979.
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
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© 1993 Springer-Verlag
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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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