Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Uniqueness of Solutions in Mean Field Games with Several Populations and Neumann Conditions Read chapter Read first chapter

2018 | OriginalPaper | Chapter

Asymptotic Methods in Regularity Theory for Nonlinear Elliptic Equations: A Survey

Authors : Edgard A. Pimentel, Makson S. Santos

Published in: PDE Models for Multi-Agent Phenomena

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

We survey recent asymptotic methods introduced in regularity theory for fully nonlinear elliptic equations. Our presentation focuses mainly on the recession function. We detail the role of this class of techniques through examples and results. Our applications include regularity in Sobolev and Hölder spaces. In addition, we produce a density result and examine ellipticity-invariant quantities, such as the Escauriaza’s exponent.

Dont have a licence yet? Then find out more about our products and how to get one now:

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!

inform now

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!

inform now

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!

inform now
previous chapter A Gradient Flow Perspective on the Quantization Problem
next chapter A Fully-Discrete Scheme for Systems of Nonlinear Fokker-Planck-Kolmogorov Equations
Literature
1.
2.
Caffarelli, L.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. 130(1), 189–213 (1989)MathSciNetCrossRef
3.
Caffarelli, L., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, vol. 43. American Mathematical Society, Providence (1995)
4.
Castillo, R., Pimentel, E.: Interior Sobolev regularity for fully nonlinear parabolic equations. Calc. Var. Partial Differ. Equ. 56, 127 (2017)MathSciNetCrossRef
5.
Escauriaza, L.: W 2, n a priori estimates for solutions to fully nonlinear equations. Indiana Univ. Math. J. 42(2), 413–423 (1993)MathSciNetCrossRef
6.
Evans, L.C.: Classical solutions of fully nonlinear, convex, second-order elliptic equations. Commun. Pure Appl. Math. 35(3), 333–363 (1982)MathSciNetCrossRef
7.
Fabes, E., Stroock, D.: The L p-integrability of Green’s functions and fundamental solutions for elliptic and parabolic equations. Duke Math. J. 51(4), 997–1016 (1984)MathSciNetCrossRef
8.
Fleming, W., Souganidis, P.: On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38(2), 293–314 (1989)MathSciNetCrossRef
9.
Krylov, N.: Estimates for derivatives of the solutions of nonlinear parabolic equations. Dokl. Akad. Nauk SSSR 274(1), 23–26 (1984)MathSciNet
10.
Krylov, N., Safonov, M.: An estimate for the probability of a diffusion process hitting a set of positive measure. Dokl. Akad. Nauk SSSR 245(1), 18–20 (1979)MathSciNet
11.
Krylov, N., Safonov, M.: A property of the solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk SSSR Ser. Mat. 44(1), 161–175, 239 (1980)
12.
Li, D., Zhang, K.: W 2, p interior estimates of fully nonlinear elliptic equations. Bull. Lond. Math. Soc. 47(2), 301–314 (2015)MathSciNetCrossRef
13.
Lin, F.: Second derivative L p-estimates for elliptic equations of nondivergent type. Proc. Am. Math. Soc. 96(3), 447–451 (1986)MathSciNetMATH
14.
Nadirashvili, N., Vladut, S.: Nonclassical solutions of fully nonlinear elliptic equations. Geom. Funct. Anal. 17(4), 1283–1296 (2007)MathSciNetCrossRef
15.
Nadirashvili, N., Vladut, S.: Singular viscosity solutions to fully nonlinear elliptic equations. J. Math. Pures Appl. (9) 89(2), 107–113 (2008)MathSciNetCrossRef
16.
Nadirashvili, N., Vladut, S.: Singular solutions of Hessian fully nonlinear elliptic equations. Adv. Math. 228(3), 1718–1741 (2011)MathSciNetCrossRef
17.
Pimentel, E.: Regularity theory for the Isaacs equation through approximation methods. Ann. Inst. H. Poincaré C Anal. Non Linéaire (2018, to appear)
18.
Pimentel, E., Teixeira, E.: Sharp Hessian integrability estimates for nonlinear elliptic equations: an asymptotic approach. J. Math. Pures Appl. 106(4), 744–767 (2016)MathSciNetCrossRef
19.
Silvestre, L., Teixeira, E.: Regularity estimates for fully non linear elliptic equations which are asymptotically convex. In: Contributions to Nonlinear Elliptic Equations and Systems, pp. 425–438. Springer, Berlin (2015)CrossRef
20.
Świȩch, A.: Another approach to the existence of value functions of stochastic differential games. J. Math. Anal. Appl. 204(3), 884–897 (1996)
21.
Świȩch, A.: W 1, p-interior estimates for solutions of fully nonlinear, uniformly elliptic equations. Adv. Diff. Equ. 2(6), 1005–1027 (1997)
22.
Teixeira, E.: Universal moduli of continuity for solutions to fully nonlinear elliptic equations. Arch. Ration. Mech. Anal. 211(3), 911–927 (2014)MathSciNetCrossRef
23.
Winter, N.: W 2, p and W 1, p-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations. Z. Anal. Anwend. 28(2), 129–164 (2009)MathSciNetCrossRef
Metadata
Title
Asymptotic Methods in Regularity Theory for Nonlinear Elliptic Equations: A Survey
Authors
Edgard A. Pimentel
Makson S. Santos
Copyright Year
2018
Book
PDE Models for Multi-Agent Phenomena

Print ISBN: 978-3-030-01946-4

Electronic ISBN: 978-3-030-01947-1

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-030-01947-1_8