Narrow Domains and the Harnack Inequality for Elliptic Equations

Safonov, M.

doi:10.1090/spmj/1401

Narrow Domains and the Harnack Inequality for Elliptic Equations
HTML articles powered by AMS MathViewer

by M. V. Safonov

St. Petersburg Math. J. 27 (2016), 509-522

DOI: https://doi.org/10.1090/spmj/1401

Published electronically: March 30, 2016

PDF | Request permission

Abstract:

We present a direct proof of Moser’s Harnack inequality that does not involve iterations. The method is based on a recursive estimate for solutions in domains of small measure. Such estimates can also be useful for other applications.

References

A. D. Aleksandrov, Uniqueness conditions and bounds for the solution of the Dirichlet problem, Vestnik Leningrad. Univ. Ser. Mat. Meh. Astronom. 18 (1963), no. 3, 5–29 (Russian, with English summary). MR 0164135
Ennio De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 (1957), 25–43 (Italian). MR 0093649
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
E. Ferretti and M. V. Safonov, Growth theorems and Harnack inequality for second order parabolic equations, Harmonic analysis and boundary value problems (Fayetteville, AR, 2000) Contemp. Math., vol. 277, Amer. Math. Soc., Providence, RI, 2001, pp. 87–112. MR 1840429, DOI 10.1090/conm/277/04540
Qing Han and Fanghua Lin, Elliptic partial differential equations, Courant Lecture Notes in Mathematics, vol. 1, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1997. MR 1669352
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
N. V. Krylov, Nelineĭnye èllipticheskie i parabolicheskie uravneniya vtorogo poryadka, “Nauka”, Moscow, 1985 (Russian). MR 815513
N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161–175, 239 (Russian). MR 563790
E. M. Landis, Second order equations of elliptic and parabolic type, Translations of Mathematical Monographs, vol. 171, American Mathematical Society, Providence, RI, 1998. Translated from the 1971 Russian original by Tamara Rozhkovskaya; With a preface by Nina Ural′tseva. MR 1487894, DOI 10.1090/mmono/171
O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
Jürgen Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468. MR 170091, DOI 10.1002/cpa.3160130308
Jürgen Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. MR 159138, DOI 10.1002/cpa.3160140329
Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 159139, DOI 10.1002/cpa.3160170106
M. V. Safonov, Harnack’s inequality for elliptic equations and Hölder property of their solutions, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 96 (1980), 272–287, 312 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 12. MR 579490