American Institute of Mathematical Sciences

Advanced
January  2009, 11(1): 87-101. doi: 10.3934/dcdsb.2009.11.87

Regularity and selecting principles for implicit ordinary differential equations

Bernard Dacorogna 1, and  Alessandro Ferriero 2,

1. 

Department of Mathématics, EPFL, 1015 Lausanne
2. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco,28049 Madrid, Spain

Received  October 2007 Revised  April 2008 Published  November 2008

Implicit Ordinary or Partial Differential Equations have been widely studied in recent times, essentially from the existence of solutions point of view. One of the main issues is to select a meaningful solution among the infinitely many ones. The most celebrated principle is the viscosity method. This selection principle is well adapted to convex Hamiltonians, but it is not always applicable to the non-convex setting.
In this work we present an alternative selecting principle that singles out the most regular solutions (which do not always coincide with the viscosity ones). Our method is based on a general regularity theorem for Implicit ODEs. We also provide several examples.
Keywords: Regularity, Implicit ODEs, Differential Inclusions, Viscosity Solutions.
Mathematics Subject Classification: Primary: 34A60, 34H05, 35F30, 49J45, 49L25.
Citation: Bernard Dacorogna, Alessandro Ferriero. Regularity and selecting principles for implicit ordinary differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 87-101. doi: 10.3934/dcdsb.2009.11.87
[1]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. Periodic solutions for implicit evolution inclusions. Evolution Equations and Control Theory, 2019, 8 (3) : 621-631. doi: 10.3934/eect.2019029
[2]

Mariusz Michta. On solutions to stochastic differential inclusions. Conference Publications, 2003, 2003 (Special) : 618-622. doi: 10.3934/proc.2003.2003.618
[3]

Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409
[4]

Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258
[5]

Mieczysław Cichoń, Bianca Satco. On the properties of solutions set for measure driven differential inclusions. Conference Publications, 2015, 2015 (special) : 287-296. doi: 10.3934/proc.2015.0287
[6]

Yejuan Wang, Tongtong Liang. Mild solutions to the time fractional Navier-Stokes delay differential inclusions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3713-3740. doi: 10.3934/dcdsb.2018312
[7]

Flaviano Battelli, Michal Fečkan. Blue sky-like catastrophe for reversible nonlinear implicit ODEs. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 895-922. doi: 10.3934/dcdss.2016034
[8]

Thomas Lorenz. Mutational inclusions: Differential inclusions in metric spaces. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 629-654. doi: 10.3934/dcdsb.2010.14.629
[9]

Pablo Ochoa, Julio Alejo Ruiz. A study of comparison, existence and regularity of viscosity and weak solutions for quasilinear equations in the Heisenberg group. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1091-1115. doi: 10.3934/cpaa.2019053
[10]

Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455
[11]

Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181
[12]

Andrej V. Plotnikov, Tatyana A. Komleva, Liliya I. Plotnikova. The averaging of fuzzy hyperbolic differential inclusions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1987-1998. doi: 10.3934/dcdsb.2017117
[13]

Guglielmo Feltrin. Positive subharmonic solutions to superlinear ODEs with indefinite weight. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 257-277. doi: 10.3934/dcdss.2018014
[14]

Farid Tari. Two-parameter families of implicit differential equations. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 139-162. doi: 10.3934/dcds.2005.13.139
[15]

Farid Tari. Geometric properties of the integral curves of an implicit differential equation. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 349-364. doi: 10.3934/dcds.2007.17.349
[16]

Kenneth Kuttler. Measurable solutions for elliptic and evolution inclusions. Evolution Equations and Control Theory, 2020, 9 (4) : 1041-1055. doi: 10.3934/eect.2020041
[17]

Kevin T. Andrews, Kenneth L. Kuttler, Ji Li, Meir Shillor. Measurable solutions to general evolution inclusions. Evolution Equations and Control Theory, 2020, 9 (4) : 935-960. doi: 10.3934/eect.2020055
[18]

Piermarco Cannarsa, Peter R. Wolenski. Semiconcavity of the value function for a class of differential inclusions. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 453-466. doi: 10.3934/dcds.2011.29.453
[19]

Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070
[20]

Tomás Caraballo, José A. Langa, José Valero. Stabilisation of differential inclusions and PDEs without uniqueness by noise. Communications on Pure and Applied Analysis, 2008, 7 (6) : 1375-1392. doi: 10.3934/cpaa.2008.7.1375

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (54)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy