Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Real analytic solutions of parabolic equations with time-measurable coefficients

Author: Jay Kovats
Journal: Proc. Amer. Math. Soc. 130 (2002), 1055-1064
MSC (1991): Primary 35B65, 35K10
Published electronically: September 14, 2001
MathSciNet review: 1873779
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We use Bernstein's technique to show that for any fixed $t$, strong solutions $u(t,x)$ of the uniformly parabolic equation $Lu:=a^{ij}(t)u_{x_{i}x_{j}}-u_{t}=0$ in $Q$ are real analytic in $Q(t)=\{x:(t,x)\in Q\}$. Here, $Q\subset \mathbb{R}^{d+1}$ is a bounded domain and the coefficients $a^{ij}(t)$ are measurable. We also use Bernstein's technique to obtain interior estimates for pure second derivatives of solutions of the fully nonlinear, uniformly parabolic, concave equation $F(D^{2}u,t)-u_{t}=0$ in $Q$, where $F$ is measurable in $t$.

References [Enhancements On Off] (What's this?)

  • [B] S.N. Bernstein, The Boundedness on the Moduli of a Sequence of Derivatives of Solutions of Equations of Parabolic Type, vol. 18, Dokl. Acad. Nauk SSSR, 1938, pp. 385-388 (Russian).
  • [Br] A. Brandt, Interior Schauder estimates for parabolic differential- (or difference-) equations via the maximum principle, Israel J. Math. 7 (1969), 254–262. MR 249803,
  • [C] S. Campanato, Proprietà di una Famiglia di Spazi Functionali, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 137-160.
  • [CC] Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007
  • [E] Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
  • [G] Mariano Giaquinta, Introduction to regularity theory for nonlinear elliptic systems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1993. MR 1239172
  • [GT] 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
  • [K] Jay Kovats, Fully nonlinear elliptic equations and the Dini condition, Comm. Partial Differential Equations 22 (1997), no. 11-12, 1911–1927. MR 1629506,
  • [K1] N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR 901759
  • [K2] N. V. Krylov, Lectures on elliptic and parabolic equations in Hölder spaces, Graduate Studies in Mathematics, vol. 12, American Mathematical Society, Providence, RI, 1996. MR 1406091
  • [K3] -, Boundedly Nonhomogeneous Elliptic and Parabolic Equations, vol. 20, Izv. Acad. Nauk., 1983, pp. 459-492, English transl. in Math. USSR Izv.
  • [KS] N.V. Krylov and M.V. Safonov, Certain Properties of Solutions of Parabolic Equations with Measurable Coefficients, vol. 16, Izv. Acad. Nauk., 1981, pp. 155-164, English transl. in Math. USSR Izv.
  • [La] 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
  • [L] Gary M. Lieberman, Intermediate Schauder theory for second order parabolic equations. IV. Time irregularity and regularity, Differential Integral Equations 5 (1992), no. 6, 1219–1236. MR 1184023
  • [LSU] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N Ural'tzeva, Linear and Quasilinear Equations of Parabolic Type, vol. 23, Amer. Math. Soc., Providence, R.I., 1968, English transl. in Translations of Math. Monographs.
  • [M] V. P. Mikhaĭlov, Partial differential equations, “Mir”, Moscow; distributed by Imported Publications, Inc., Chicago, Ill., 1978. Translated from the Russian by P. C. Sinha. MR 601389
  • [W] Lihe Wang, On the regularity theory of fully nonlinear parabolic equations. II, Comm. Pure Appl. Math. 45 (1992), no. 2, 141–178. MR 1139064,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35B65, 35K10

Retrieve articles in all journals with MSC (1991): 35B65, 35K10

Additional Information

Jay Kovats
Affiliation: Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida 32901

Received by editor(s): October 4, 2000
Published electronically: September 14, 2001
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2001 American Mathematical Society