The Harnack inequality and related properties for solutions of elliptic and parabolic equations with divergence-free lower-order coefficients
HTML articles powered by AMS MathViewer
- by
A. I. Nazarov and N. N. Ural′tseva
Translated by: the authors - St. Petersburg Math. J. 23 (2012), 93-115
- DOI: https://doi.org/10.1090/S1061-0022-2011-01188-4
- Published electronically: November 8, 2011
- PDF | Request permission
Abstract:
The paper is devoted to the question as to how “bad” the junior coefficients of elliptic and parabolic equations may be in order that classical properties of their solutions (such as the strict maximum principle, the Harnack inequality and the Liouville theorem) still occur. The answers are given in terms of the Lebesgue and Morrey spaces.References
- 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
- Charles B. Morrey Jr., Second order elliptic equations in several variables and Hölder continuity, Math. Z. 72 (1959/1960), 146–164. MR 0120446, DOI 10.1007/BF01162944
- John Nash, Parabolic equations, Proc. Nat. Acad. Sci. U.S.A. 43 (1957), 754–758. MR 89986, DOI 10.1073/pnas.43.8.754
- O. A. Ladyženskaja and N. N. Ural′ceva, A boundary-value problem for linear and quasi-linear parabolic equations, Dokl. Akad. Nauk SSSR 139 (1961), 544–547 (Russian). MR 0141891
- 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
- A. I. Nazarov and N. N. Ural′tseva, Qualitative properties of solutions to elliptic and parabolic equations with unbounded lower-order coefficients, SPbMS El. Prepr. Archive. N 2009-05, 6 p.
- Qi S. Zhang, A strong regularity result for parabolic equations, Comm. Math. Phys. 244 (2004), no. 2, 245–260. MR 2031029, DOI 10.1007/s00220-003-0974-6
- Gabriel Koch, Nikolai Nadirashvili, Gregory A. Seregin, and Vladimir Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math. 203 (2009), no. 1, 83–105. MR 2545826, DOI 10.1007/s11511-009-0039-6
- Chiun-Chuan Chen, Robert M. Strain, Horng-Tzer Yau, and Tai-Peng Tsai, Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations, Int. Math. Res. Not. IMRN 9 (2008), Art. ID rnn016, 31. MR 2429247, DOI 10.1093/imrn/rnn016
- G. Seregin, L. Silvestre, V. Šverák, and A. Zlatos, On divergence-free drifts, Preprint arXiv:1010.6025v1.
- Neil S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 27 (1973), 265–308. MR 369884
- Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184, DOI 10.1142/3302
- O. A. Ladyzhenskaya and N. N. Ural′tseva, Lineĭnye i kvazilineĭnye uravneniya èllipticheskogo tipa, Izdat. “Nauka”, Moscow, 1973 (Russian). Second edition, revised. MR 0509265
- 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
- Elliott H. Lieb and Michael Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 1997. MR 1415616, DOI 10.1090/gsm/014
- Mikhail V. Safonov, Mean value theorems and Harnack inequalities for second-order parabolic equations, Nonlinear problems in mathematical physics and related topics, II, Int. Math. Ser. (N. Y.), vol. 2, Kluwer/Plenum, New York, 2002, pp. 329–352. MR 1972004, DOI 10.1007/978-1-4615-0701-7_{1}8
- V. G. Maz′ya and I. E. Verbitsky, Form boundedness of the general second-order differential operator, Comm. Pure Appl. Math. 59 (2006), no. 9, 1286–1329. MR 2237288, DOI 10.1002/cpa.20122
- Giovanni Maria Troianiello, Elliptic differential equations and obstacle problems, The University Series in Mathematics, Plenum Press, New York, 1987. MR 1094820, DOI 10.1007/978-1-4899-3614-1
- O. V. Besov, V. P. Il′in, and S. M. Nikol′skiĭ, Integral′nye predstavleniya funktsiĭ i teoremy vlozheniya, 2nd ed., Fizmatlit “Nauka”, Moscow, 1996 (Russian, with Russian summary). MR 1450401
- M. V. Safonov, Non-divergence elliptic equations of second order with unbounded drift, Nonlinear partial differential equations and related topics, Amer. Math. Soc. Transl. Ser. 2, vol. 229, Amer. Math. Soc., Providence, RI, 2010, pp. 211–232. MR 2667641, DOI 10.1090/trans2/229/13
Bibliographic Information
- A. I. Nazarov
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya Ul. 28, Stary Petergof, St. Petersburg 198504, Russia
- MR Author ID: 228194
- Email: al.il.nazarov@gmail.com
- N. N. Ural′tseva
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya Ul. 28, Stary Petergof, St. Petersburg 198504, Russia
- Email: uunur@NU1253.spb.edu
- Received by editor(s): October 12, 2010
- Published electronically: November 8, 2011
- Additional Notes: Partially supported by RFBR (grant no. 08-01-00748) and by grant NSh.4210.2010.1.
- © Copyright 2011 American Mathematical Society
- Journal: St. Petersburg Math. J. 23 (2012), 93-115
- MSC (2010): Primary 35B50, 35B53, 35B45
- DOI: https://doi.org/10.1090/S1061-0022-2011-01188-4
- MathSciNet review: 2760150
Dedicated: To the memory of Mikhail Solomonovich Birman