Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Elliptic equations with VMO a, b$\,\in L_{d}$, and c$\,\in L_{d/2}$
HTML articles powered by AMS MathViewer

by N. V. Krylov PDF
Trans. Amer. Math. Soc. 374 (2021), 2805-2822 Request permission


We consider elliptic equations with operators $L=a^{ij}D_{ij}+b^{i}D_{i}-c$ with $a$ being almost in VMO, $b\in L_{d}$ and $c\in L_{q}$, $c\geq 0$, $d>q\geq d/2$. We prove the solvability of $Lu=f\in L_{p}$ in bounded $C^{1,1}$-domains, $1<p\leq q$, and of $\lambda u-Lu=f$ in the whole space for any $\lambda >0$. Weak uniqueness of the martingale problem associated with such operators is also obtained.
  • D. E. Apushkinskaya, A. I. Nazarov, D. K. Palagachev, and L. G. Softova, Venttsel boundary value problems with discontinuous data, arXiv:1907.03017.
  • Lisa Beck, Franco Flandoli, Massimiliano Gubinelli, and Mario Maurelli, Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness, Electron. J. Probab. 24 (2019), Paper No. 136, 72. MR 4040996, DOI 10.1214/19-ejp379
  • Hongjie Dong and Doyoon Kim, On the $L_p$-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arch. Ration. Mech. Anal. 199 (2011), no. 3, 889–941. MR 2771670, DOI 10.1007/s00205-010-0345-3
  • Byungsoo Kang and Hyunseok Kim, On $L^p$-resolvent estimates for second-order elliptic equations in divergence form, Potential Anal. 50 (2019), no. 1, 107–133. MR 3900848, DOI 10.1007/s11118-017-9675-1
  • N. V. Krylov, Controlled diffusion processes, Applications of Mathematics, vol. 14, Springer-Verlag, New York-Berlin, 1980. Translated from the Russian by A. B. Aries. MR 601776
  • N. V. Krylov, Introduction to the theory of random processes, Graduate Studies in Mathematics, vol. 43, American Mathematical Society, Providence, RI, 2002. MR 1885884, DOI 10.1090/gsm/043
  • N. V. Krylov, On weak uniqueness for some diffusions with discontinuous coefficients, Stochastic Process. Appl. 113 (2004), no. 1, 37–64. MR 2078536, DOI 10.1016/
  • N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Mathematics, vol. 96, American Mathematical Society, Providence, RI, 2008. MR 2435520, DOI 10.1090/gsm/096
  • N. V. Krylov, On stochastic Itô processes with drift in $L_{d}$, arXiv:2001.03660.
  • N. V. Krylov, On stochastic equations with drift in $L_{d}$, arXiv:2001.04008.
  • N.V. Krylov, On diffusion processes with drift in $L_{d}$, arXiv:2001.04950.
  • O. A. Ladyzhenskaya and N. N. Ural′tseva, Lineĭnye i kvazilineĭnye uravneniya èllipticheskogo tipa, Izdat. “Nauka”, Moscow, 1973 (Russian). Second edition, revised. MR 0509265
  • Vladimir G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR 817985, DOI 10.1007/978-3-662-09922-3
  • 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
  • Guido Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 189–258 (French). MR 192177
  • Neil S. Trudinger, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 27 (1973), 265–308. MR 369884
  • Saisai Yang and Tusheng Zhang, Dirichlet boundary value problems for elliptic operators with measure data: a probabilistic approach, arXiv:1804.01819
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 35K10, 35J15, 60J60
  • Retrieve articles in all journals with MSC (2020): 35K10, 35J15, 60J60
Additional Information
  • N. V. Krylov
  • Affiliation: Department of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, Minnesota 55455
  • MR Author ID: 189683
  • Email:
  • Received by editor(s): March 24, 2020
  • Received by editor(s) in revised form: August 8, 2020
  • Published electronically: January 20, 2021
  • © Copyright 2021 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 374 (2021), 2805-2822
  • MSC (2020): Primary 35K10, 35J15, 60J60
  • DOI:
  • MathSciNet review: 4223034