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 Mat. Sb. (N.S.), 1986, Volume 130(172), Number 2(6), Pages 207–221 (Mi msb1865)

On estimates of the maximum of a solution of a parabolic equation and estimates of the distribution of a semimartingale

N. V. Krylov

Abstract: Estimates are proved for the maximum of a solution of a linear parabolic equation in terms of the $\mathscr L_p$-norm of the right-hand side. The coefficients of the first derivatives are assumed to be integrable to a suitable power. Various boundary value problems are considered. Corresponding $\mathscr L_p$-estimates are proved also for the distributions of semimartingales.
Bibliography: 16 titles.

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English version:
Mathematics of the USSR-Sbornik, 1987, 58:1, 207–221

Bibliographic databases:

UDC: 517.95
MSC: Primary 35K20, 60G48; Secondary 35B50, 60E99

Citation: N. V. Krylov, “On estimates of the maximum of a solution of a parabolic equation and estimates of the distribution of a semimartingale”, Mat. Sb. (N.S.), 130(172):2(6) (1986), 207–221; Math. USSR-Sb., 58:1 (1987), 207–221

Citation in format AMSBIB
\Bibitem{Kry86} \by N.~V.~Krylov \paper On estimates of the maximum of a~solution of a~parabolic equation and estimates of the distribution of a~semimartingale \jour Mat. Sb. (N.S.) \yr 1986 \vol 130(172) \issue 2(6) \pages 207--221 \mathnet{http://mi.mathnet.ru/msb1865} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=854972} \zmath{https://zbmath.org/?q=an:0625.35041} \transl \jour Math. USSR-Sb. \yr 1987 \vol 58 \issue 1 \pages 207--221 \crossref{https://doi.org/10.1070/SM1987v058n01ABEH003100} 

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This publication is cited in the following articles:
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2. Lieberman C.M., “The Maximum Principle for Equations with Composite Coefficients”, Electron. J. Differ. Equ., 2000, 38
3. Gyongy I., Martinez T., “On Stochastic Differential Equations with Locally Unbounded Drift”, Czech. Math. J., 51:4 (2001), 763–783
4. Zhang X., “Strong Solutions of SDEs with Singular Drift and Sobolev Diffusion Coefficients”, Stoch. Process. Their Appl., 115:11 (2005), 1805–1818
5. Sergei B. Kuksin, “On Distribution of Energy and Vorticity for Solutions of 2D Navier–Stokes Equation with Small Viscosity”, Comm Math Phys, 2008
6. Kurenok V.P., Lepeyev A.N., “On Multi-Dimensional SDEs with Locally Integrable Coefficients”, Rocky Mt. J. Math., 38:1 (2008), 139–174
7. Meyer-Brandis T., Proske F., “Construction of Strong Solutions of SDEs via Malliavin Calculus”, J. Funct. Anal., 258:11 (2010), 3922–3953
8. Zhang X., “Stochastic Homeomorphism Flows of SDEs with Singular Drifts and Sobolev Diffusion Coefficients”, Electron. J. Probab., 16 (2011), 38, 1096–1116
9. Nazarov A.I., “A Centennial of the Zaremba-Hopf-Oleinik Lemma”, SIAM J. Math. Anal., 44:1 (2012), 437–453
10. Krylov N.V., “An Ersatz Existence Theorem for Fully Nonlinear Parabolic Equations Without Convexity Assumptions”, SIAM J. Math. Anal., 45:6 (2013), 3331–3359
11. Chen G., “Non-divergence parabolic equations of second order with critical drift in Lebesgue spaces”, J. Differ. Equ., 262:3, 2 (2017), 2414–2448
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