Real analytic solutions of parabolic equations with timemeasurable coefficients
Author:
Jay Kovats
Journal:
Proc. Amer. Math. Soc. 130 (2002), 10551064
MSC (1991):
Primary 35B65, 35K10
Published electronically:
September 14, 2001
MathSciNet review:
1873779
Fulltext PDF Free Access
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Abstract: We use Bernstein's technique to show that for any fixed , strong solutions of the uniformly parabolic equation in are real analytic in . Here, is a bounded domain and the coefficients 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 in , where is measurable in .
 [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. 385388 (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 0249803
(40 #3044)
 [C]
S. Campanato, Proprietà di una Famiglia di Spazi Functionali, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 137160.
 [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
(96h:35046)
 [E]
Lawrence
C. Evans, Partial differential equations, Graduate Studies in
Mathematics, vol. 19, American Mathematical Society, Providence, RI,
1998. MR
1625845 (99e:35001)
 [G]
Mariano
Giaquinta, Introduction to regularity theory for nonlinear elliptic
systems, Lectures in Mathematics ETH Zürich, Birkhäuser
Verlag, Basel, 1993. MR 1239172
(94g:49002)
 [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,
SpringerVerlag, Berlin, 1983. MR 737190
(86c:35035)
 [K]
Jay
Kovats, Fully nonlinear elliptic equations and the Dini
condition, Comm. Partial Differential Equations 22
(1997), no. 1112, 1911–1927. MR 1629506
(99h:35048), http://dx.doi.org/10.1080/03605309708821325
 [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
(88d:35005)
 [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
(97i:35001)
 [K3]
, Boundedly Nonhomogeneous Elliptic and Parabolic Equations, vol. 20, Izv. Acad. Nauk., 1983, pp. 459492, 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. 155164, 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
(98k:35034)
 [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
(93i:35068)
 [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, 1978. Translated from the Russian by P. C.
Sinha. MR
601389 (82a:35003a)
 [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
(92m:35127), http://dx.doi.org/10.1002/cpa.3160450202
 [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. 385388 (Russian).
 [Br]
 A. Brandt, Interior Schauder Estimates for Parabolic Differential (or Difference) Equations via the Maximum Principle, vol. 7, Israel J. Math., 1969, pp. 254262. MR 40:3044
 [C]
 S. Campanato, Proprietà di una Famiglia di Spazi Functionali, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 137160.
 [CC]
 L. Caffarelli and X. Cabre, Fully Nonlinear Elliptic Equations, Amer. Math. Soc., Providence, R.I., 1995. MR 96h:35046
 [E]
 L.C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence, R.I., 1998. MR 99e:35001
 [G]
 M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Birkhäuser Verlag, Basel, 1993. MR 94g:49002
 [GT]
 D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., SpringerVerlag, BerlinHeidelbergNew YorkTokyo, 1983. MR 86c:35035
 [K]
 J. Kovats, Fully Nonlinear Elliptic Equations and the Dini Condition, Communications in PDE 22 (1112) (1997), 19111927. MR 99h:35048
 [K1]
 N.V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order Equations, Nauka, Moscow, 1987, English transl. Reidel, Dordrecht. MR 88d:35005
 [K2]
 , Lectures on Elliptic and Parabolic Equations in Hölder Spaces, Amer. Math. Soc., Providence, R.I., 1996. MR 97i:35001
 [K3]
 , Boundedly Nonhomogeneous Elliptic and Parabolic Equations, vol. 20, Izv. Acad. Nauk., 1983, pp. 459492, 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. 155164, English transl. in Math. USSR Izv.
 [La]
 E.M. Landis, Second Order Equations of Elliptic and Parabolic Type, vol. 171, Amer. Math. Soc., Providence, R.I., 1998, English transl. in Translations of Math. Monographs. MR 98k:35034
 [L]
 G.M. Lieberman, Intermediate Schauder Theory for Second Order Parabolic Equations IV. Time Irregularity and Regularity, Differential and Integral Equations 5 (1992), 12191236. MR 93i:35068
 [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. Mikhailov, Partial Differential Equations, Mir, Moscow, 1978. MR 82a:35003a
 [W]
 Wang L., On the Regularity Theory of Fully Nonlinear Parabolic Equations: II, Comm. on Pure and Applied Math. 45 (1992), 141178. MR 92m:35127
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Additional Information
Jay Kovats
Affiliation:
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida 32901
Email:
jkovats@zach.fit.edu
DOI:
http://dx.doi.org/10.1090/S0002993901061639
PII:
S 00029939(01)061639
Received by editor(s):
October 4, 2000
Published electronically:
September 14, 2001
Communicated by:
David S. Tartakoff
Article copyright:
© Copyright 2001 American Mathematical Society
