We prove the boundary Harnack principle for ratios of solutions u/v of non-divergence second order elliptic equations Lu = a ij D ij u + b i D i u = 0 in a bounded domain Ω ⊂ \( {\mathbb R} \) n. We assume that b i ∈ L n(Ω) and Ω is a twisted Hölder domain of order α ∈ (1/2, 1]. Based on this result, we derive the Hölder regularity of u/v for uniform domains. Bibliography: 27 titles.
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References
L. C. Evans, “Classical solutions of fuly nonlinear, convex, second-order elliptic equations,” Comm. Pure Appl. Math. 35, 333–363 (1982).
N. V. Krylov, “Boundedly nonhomogeneous elliptic and parabolic equations,” [in Russian] Izv. Akad. Nauk SSSR, Ser. Mat. 46, 487–523 (1982); English transl.: Math. USSR Izvestija 20, 459–492 (1983).
N. V. Krylov, “Boundedly nonhomogeneous elliptic and parabolic equations in a domain,” [in Russian] Izv. Akad. Nauk SSSR, Ser. Mat. 47, 75–108 (1983); English transl.: Math. USSR Izvestija 22, 67–97 (1984).
L. Nirenberg, “On nonlinear partial differential equations and Hölder continuity,” Comm. Pure Appl. Math. 6, 103–156 (1953).
N. V. Krylov and M. V. Safonov, “A certain property of solutions of parabolic equations with measurable coefficients,” [in Russian] Izv.Akad. Nauk SSSR, Ser. Mat. 44 161–175 (1980); English transl.: Math. USSR Izvestija 16, 151–164 (1981).
M. V. Safonov, “Harnack inequality for elliptic equations and the Hölder property of their solutions,” [in Russian] Zap. Nauchn. Semin. LOMI 96, 272–287 (1980); English transl.: J. Math. Sci., New York 21, No. 5, 851–863 (1983).
N. Nadirashvili and S. Vlăduţ, “Nonclassical solutions of fully nonlinear elliptic equations,” Geom. Funct. Anal. 17, 1283–1296 (2007).
N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of Second Order [in Russian], Nauka, Moscow (1985); English transl.: D. Reidel, Dordrecht, Holland (1987).
H. Kim and M. V. Safonov, “Carleson type estimates for second order elliptic equations with unbounded drift,” J. Math. Sci., New York 176, No. 6, 928–944 (2011).
L. Carleson, “On the existence of boundary values for harmonic functions in several variables,” Ark. Mat. 4, 339–343 (1961).
A. Ancona, “Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien,” Ann. Inst. Fourier (Grenoble) 28, 169–213 (1978).
B. Dahlberg, “Estimates for harmonic measure,” Arch. Rat. Mech. Anal. 65, 275–288 (1977).
J.-M. G. Wu, “Comparison of kernel functions, boundary Harnack principle, and relative Fatou theorem on Lipschitz domains,” Ann. Inst. Fourier (Grenoble) 28, 147–167 (1978).
L. A. Caffarelli, E. B. Fabes, S. Mortola, and S. Salsa, “Boundary behavior of nonnegative solutions of elliptic operators in divergence form,” Indiana J. Math. 30, 621–640 (1981).
P. E. Bauman, “Positive solutions of elliptic equations in non-divergence form and their adjoints,” Ark. Mat. 22, No. 2, 153–173 (1984).
C. E. Kenig, “Potential theory of non-divergence form elliptic equations. Dirichlet forms (Varenna, 1992),” Lecture Notes in Math., Springer, Berlin 1563, 89–128 (1993).
D. S. Jerison and C. E. Kenig, “Boundary behavior of harmonic functions in non-tangentially accessible domains,” Adv. Math. 46, 80–147 (1982).
R. F. Bass and K. Burdzy, “A boundary Harnack principle in twisted Hölder domain,” Ann. Math. 134, 253–276 (1991).
H. Aikawa, “Boundary Harnack principle and Martin boundary for a uniform domain,” J. Math. Soc. Japan 53. No. 1, 119–145 (2001).
R. F. Bass and K. Burdzy, “Lifetimes of conditioned diffusions,” Probab. Th. Rel. Fields 91, 405–443 (1992).
R. Bañuelos, R. F. Bass, and K. Burdzy, “Hölder domains and the boundary Harnack principle,” Duke Math. J. 64(1), 195–200 (1991).
F. Ferrari, “On boundary behavior of harmonic functions in Hölder domains,” J. Fourier Anal. Appl. 4, 447–461 (1998).
R. F. Bass and K. Burdzy, “The boundary Harnack principle for non-divergence form elliptic operators,” J. London Math. Soc. 50, 157–169 (1994).
A. D. Aleksandrov, “Uniqueness conditions and estimates for the solution of the Dirichlet problem,” [in Russian], Vestn. Leningr. Univ. 18, No. 3, 5–29 (1963); English transl.: AMS Transl. (2) 68, 89–119 (1968).
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, New York (1983).
M. V. Safonov, “Non-divergence elliptic equations of second order with unbounded drift,” In: Nonlinear Partial Differential Equations and Related Topics, AMS Transl. (2) 229, 211–232 (2010).
E. M. Landis, Second Order Equations of Elliptic and Parabolic Type [in Russian], Nauka, Moscow (1971); English transl.: Transl Math. Monographs 171, AMS, Providence, RI (1997).
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Dedicated to Nicolai Krylov
Translated from Problems in Mathematical Analysis 61, October 2011, pp. 109–122.
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Kim, H., Safonov, M. Boundary Harnack principle for second order elliptic equations with unbounded drift. J Math Sci 179, 127–143 (2011). https://doi.org/10.1007/s10958-011-0585-2
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DOI: https://doi.org/10.1007/s10958-011-0585-2