Skip to main content

2013 | OriginalPaper | Buchkapitel

2. An Introduction to Fully Nonlinear Parabolic Equations

verfasst von : Cyril Imbert, Luis Silvestre

Erschienen in: An Introduction to the Kähler-Ricci Flow

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

These notes contain a short exposition of selected results about parabolic equations: Schauder estimates for linear parabolic equations with Hölder coefficients, some existence, uniqueness and regularity results for viscosity solutions of fully nonlinear parabolic equations (including degenerate ones), the Harnack inequality for fully nonlinear uniformly parabolic equations.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Anhänge
Nur mit Berechtigung zugänglich
Literatur
[ALL97]
Zurück zum Zitat O. Alvarez, J.-M. Lasry, P.-L. Lions, Convex viscosity solutions and state constraints. J. Math. Pures Appl. (9) 76(3), 265–288 (1997) O. Alvarez, J.-M. Lasry, P.-L. Lions, Convex viscosity solutions and state constraints. J. Math. Pures Appl. (9) 76(3), 265–288 (1997)
[Barl94]
Zurück zum Zitat G. Barles, in Solutions de viscosité des équations de Hamilton-Jacobi. Mathématiques & Applications (Berlin), vol. 17 (Springer, Paris, 1994), x + 194 pp. G. Barles, in Solutions de viscosité des équations de Hamilton-Jacobi. Mathématiques & Applications (Berlin), vol. 17 (Springer, Paris, 1994), x + 194 pp.
[CafCab]
Zurück zum Zitat L.A. Caffarelli, X. Cabre, in Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, vol. 43 (American Mathematical Society, Providence, 1995), vi + 104 pp. L.A. Caffarelli, X. Cabre, in Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, vol. 43 (American Mathematical Society, Providence, 1995), vi + 104 pp.
[CanSin04]
Zurück zum Zitat P. Cannarsa, C. Sinestrari, Semiconcave functions, in Hamilton-Jacobi Equations, and Optimal Control. Progress in Nonlinear Differential Equations and Their Applications, vol. 58 (Birkhäuser, Boston, 2004), xiv + 304 pp. P. Cannarsa, C. Sinestrari, Semiconcave functions, in Hamilton-Jacobi Equations, and Optimal Control. Progress in Nonlinear Differential Equations and Their Applications, vol. 58 (Birkhäuser, Boston, 2004), xiv + 304 pp.
[CL81]
Zurück zum Zitat M.G. Crandall, P.-L. Lions, Condition d’unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre. C. R. Acad. Sci. Paris Sér. I Math. 292(3), 183–186 (1981)MathSciNetMATH M.G. Crandall, P.-L. Lions, Condition d’unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre. C. R. Acad. Sci. Paris Sér. I Math. 292(3), 183–186 (1981)MathSciNetMATH
[CIL92]
Zurück zum Zitat M.G. Crandall, H. Ishii, P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992) M.G. Crandall, H. Ishii, P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)
[Dong91]
Zurück zum Zitat G.C. Dong, in Nonlinear Partial Differential Equations of Second Order. Translated from the Chinese by Kai Seng Chou [Kaising Tso]. Translations of Mathematical Monographs, vol. 95 (American Mathematical Society, Providence, 1991), viii + 251 pp. G.C. Dong, in Nonlinear Partial Differential Equations of Second Order. Translated from the Chinese by Kai Seng Chou [Kaising Tso]. Translations of Mathematical Monographs, vol. 95 (American Mathematical Society, Providence, 1991), viii + 251 pp.
[EG92]
Zurück zum Zitat L.C. Evans, R.F. Gariepy, in Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics (CRC Press, Boca Raton, 1992), viii + 268 pp. L.C. Evans, R.F. Gariepy, in Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics (CRC Press, Boca Raton, 1992), viii + 268 pp.
[GT01]
Zurück zum Zitat D. Gilbarg, N.S. Trudinger, in Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edn. Classics in Mathematics (Springer, Berlin, 2001), xiv + 517 pp. D. Gilbarg, N.S. Trudinger, in Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edn. Classics in Mathematics (Springer, Berlin, 2001), xiv + 517 pp.
[HUL]
Zurück zum Zitat J.-B. Hirriart-Urruty, C. Lemaréchal, Fundamentals of convex analysis. Abridged version of Convex analysis and minimization algorithms. I [Springer, Berlin, 1993; MR1261420 (95m:90001)] and II [Springer, Berlin, 1993; MR1295240 (95m:90002)]. Grundlehren Text Editions (Springer, Berlin, 2001), x + 259 pp. J.-B. Hirriart-Urruty, C. Lemaréchal, Fundamentals of convex analysis. Abridged version of Convex analysis and minimization algorithms. I [Springer, Berlin, 1993; MR1261420 (95m:90001)] and II [Springer, Berlin, 1993; MR1295240 (95m:90002)]. Grundlehren Text Editions (Springer, Berlin, 2001), x + 259 pp.
[Imb06]
Zurück zum Zitat C. Imbert, Convexity of solutions and C 1, 1 estimates for fully nonlinear elliptic equations. J. Math. Pure Appl. (9) 85(6), 791–807 (2006) C. Imbert, Convexity of solutions and C 1, 1 estimates for fully nonlinear elliptic equations. J. Math. Pure Appl. (9) 85(6), 791–807 (2006)
[Ish89]
Zurück zum Zitat H. Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Comm. Pure Appl. Math. 42(1), 15–45 (1989)MathSciNetCrossRefMATH H. Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Comm. Pure Appl. Math. 42(1), 15–45 (1989)MathSciNetCrossRefMATH
[IL90]
Zurück zum Zitat H. Ishii, P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equat. 83(1), 26–78 (1990)MathSciNetCrossRefMATH H. Ishii, P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equat. 83(1), 26–78 (1990)MathSciNetCrossRefMATH
[Jens88]
Zurück zum Zitat R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Ration. Mech. Anal. 101(1), 1–27 (1988)CrossRefMATH R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Ration. Mech. Anal. 101(1), 1–27 (1988)CrossRefMATH
[Kryl76]
Zurück zum Zitat N.V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation. Sibirsk. Mat. Z. 17(2), 290–303, 478 (1976) N.V. Krylov, Sequences of convex functions, and estimates of the maximum of the solution of a parabolic equation. Sibirsk. Mat. Z. 17(2), 290–303, 478 (1976)
[Kryl87]
Zurück zum Zitat N.V. Krylov, in Nonlinear Elliptic and Parabolic Equations of the Second Order. Translated from the Russian by P.L. Buzytsky. Mathematics and Its Applications (Soviet Series), vol. 7 (D. Reidel Publishing Co., Dordrecht, 1987), xiv + 462 pp. N.V. Krylov, in Nonlinear Elliptic and Parabolic Equations of the Second Order. Translated from the Russian by P.L. Buzytsky. Mathematics and Its Applications (Soviet Series), vol. 7 (D. Reidel Publishing Co., Dordrecht, 1987), xiv + 462 pp.
[Kryl96]
Zurück zum Zitat N.V. Krylov, in Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, vol. 12 (American Mathematical Society, Providence, 1996), xii + 164 pp. N.V. Krylov, in Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, vol. 12 (American Mathematical Society, Providence, 1996), xii + 164 pp.
[Kryl97]
Zurück zum Zitat N.V. Krylov, Fully nonlinear second order elliptic equations: recent development. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(3–4), 569–595 (1997) N.V. Krylov, Fully nonlinear second order elliptic equations: recent development. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25(3–4), 569–595 (1997)
[LSU67]
Zurück zum Zitat O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva, in Linear and Quasilinear Equations of Parabolic Type (Russian). Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23 (American Mathematical Society, Providence, 1967), xi + 648 pp. O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva, in Linear and Quasilinear Equations of Parabolic Type (Russian). Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23 (American Mathematical Society, Providence, 1967), xi + 648 pp.
[Lieb96]
Zurück zum Zitat G.M. Lieberman, Second Order Parabolic Differential Equations (World Scientific, River Edge, 1996)CrossRefMATH G.M. Lieberman, Second Order Parabolic Differential Equations (World Scientific, River Edge, 1996)CrossRefMATH
[Lions83]
Zurück zum Zitat P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, II. Viscosity solutions and uniqueness. Comm. Partial Differ. Equat. 8(11), 1229–1276 (1983)CrossRefMATH P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, II. Viscosity solutions and uniqueness. Comm. Partial Differ. Equat. 8(11), 1229–1276 (1983)CrossRefMATH
[Saf84]
Zurück zum Zitat M.V. Safonov, The classical solution of the elliptic Bellman equation (Russian). Dokl. Akad. Nauk SSSR 278(4), 810–813 (1984)MathSciNet M.V. Safonov, The classical solution of the elliptic Bellman equation (Russian). Dokl. Akad. Nauk SSSR 278(4), 810–813 (1984)MathSciNet
[Tso85]
Zurück zum Zitat K. Tso, On an Aleksandrov-Bakelman type maximum principle for second-order parabolic equations. Comm. Partial Differ. Equat. 10(5), 543–553 (1985)MathSciNetCrossRefMATH K. Tso, On an Aleksandrov-Bakelman type maximum principle for second-order parabolic equations. Comm. Partial Differ. Equat. 10(5), 543–553 (1985)MathSciNetCrossRefMATH
[Wang92a]
Zurück zum Zitat L. Wang, On the regularity theory of fully nonlinear parabolic equations, I. Comm. Pure Appl. Math. 45(1), 27–76 (1992)CrossRefMATH L. Wang, On the regularity theory of fully nonlinear parabolic equations, I. Comm. Pure Appl. Math. 45(1), 27–76 (1992)CrossRefMATH
Metadaten
Titel
An Introduction to Fully Nonlinear Parabolic Equations
verfasst von
Cyril Imbert
Luis Silvestre
Copyright-Jahr
2013
DOI
https://doi.org/10.1007/978-3-319-00819-6_2