American Institute of Mathematical Sciences

Advanced
December  2020, 9(4): 1041-1055. doi: 10.3934/eect.2020041

Measurable solutions for elliptic and evolution inclusions

Kenneth Kuttler ,

249 Havenside Court, Grantsville, Utah 84029, USA

* Kenneth Kuttler

I would like to thank the anonymous referees for finding some loose ends and things which needed improvement.

Received  August 2019 Revised  December 2019 Published  March 2020

This paper obtains existence of random variable solutions to elliptic and evolution inclusions. As a special case, surprising theorems are obtained for the quasistatic problems. A new existence theorem is also presented for evolution inclusions with set valued operators dependent on elements of a measurable space.

Keywords: Measurable quasistatic inclusions, measurable evolution inclusions.
Mathematics Subject Classification: Primary: 35R60; Secondary: 35Q74, 60H25.
Citation: Kenneth Kuttler. Measurable solutions for elliptic and evolution inclusions. Evolution Equations & Control Theory, 2020, 9 (4) : 1041-1055. doi: 10.3934/eect.2020041
References:
[1]

K. T. Andrews, K. L. Kuttler, J. Li and M. Shillor, Measurable solutions for elliptic inclusions and quasistatic problems, Comput. Math. Appl., 77 (2019), 2869-2882.  doi: 10.1016/j.camwa.2018.09.025.  Google Scholar

[2]

A. Bensoussan and R. Temam, Équations stochastiques du type Navier-Stokes, J. Functional Analysis, 13 (1973), 195-222.  doi: 10.1016/0022-1236(73)90045-1.  Google Scholar

[3]

H. Brezis, Équations et inéquations non lin éaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175.  doi: 10.5802/aif.280.  Google Scholar

[4]

Z. Denkowski, S. Migórski and N. S. Papageorgiu, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic Publishers, Boston, MA, 2003; Somerville, MA, (2002). doi: 10.1007/978-1-4419-9158-4.  Google Scholar

[5]

W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Vol. 30, Studies in Advanced Mathematics, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002.  Google Scholar

[6]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I. Theory. Mathematics and its Applications, Vol. 419, Kluwer Academic Publishers, Dordrecht, 1997.  Google Scholar

[7]

K. L. Kuttler, J. Li and M. Shillor, A general product measurability theorem with applications to variational inequalities, Electron. J. Differential Equations, 90 (2016), 12 pp.  Google Scholar

[8]

K. L. Kuttler and M. Shillor, Product measurability with applications to a stochastic contact problem with friction, Electron. J. Differential Equations, 258 (2014), 29 pp.  Google Scholar

[9]

K. Kuttler, Non-degenerate implicit evolution inclusions, Electron. J. Differential Equations, 34 (2000), 20 pp.  Google Scholar

[10]

K. L. Kuttler and M. Shillor, Set-valued pseudomonotone maps and degenerate evolution inclusions, Commun. Contemp. Math., 1 (1999), 87-123.  doi: 10.1142/S0219199799000067.  Google Scholar

[11]

J.-L. Lions, Quelques Méthods de Résolution des Problèmes aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[12]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Vol. 26, Advances in Mechanics and Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar

[13]

M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact, Vol. 655, Lecture Notes in Physics, Springer, Berlin, Heidelberg, 2004. doi: 10.1007/b99799.  Google Scholar

[14]

M.Sofonea, W. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Vol. 276, Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar

show all references

References:
[1]

K. T. Andrews, K. L. Kuttler, J. Li and M. Shillor, Measurable solutions for elliptic inclusions and quasistatic problems, Comput. Math. Appl., 77 (2019), 2869-2882.  doi: 10.1016/j.camwa.2018.09.025.  Google Scholar

[2]

A. Bensoussan and R. Temam, Équations stochastiques du type Navier-Stokes, J. Functional Analysis, 13 (1973), 195-222.  doi: 10.1016/0022-1236(73)90045-1.  Google Scholar

[3]

H. Brezis, Équations et inéquations non lin éaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18 (1968), 115-175.  doi: 10.5802/aif.280.  Google Scholar

[4]

Z. Denkowski, S. Migórski and N. S. Papageorgiu, An Introduction to Nonlinear Analysis: Theory, Kluwer Academic Publishers, Boston, MA, 2003; Somerville, MA, (2002). doi: 10.1007/978-1-4419-9158-4.  Google Scholar

[5]

W. Han and M. Sofonea, Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Vol. 30, Studies in Advanced Mathematics, American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2002.  Google Scholar

[6]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Vol. I. Theory. Mathematics and its Applications, Vol. 419, Kluwer Academic Publishers, Dordrecht, 1997.  Google Scholar

[7]

K. L. Kuttler, J. Li and M. Shillor, A general product measurability theorem with applications to variational inequalities, Electron. J. Differential Equations, 90 (2016), 12 pp.  Google Scholar

[8]

K. L. Kuttler and M. Shillor, Product measurability with applications to a stochastic contact problem with friction, Electron. J. Differential Equations, 258 (2014), 29 pp.  Google Scholar

[9]

K. Kuttler, Non-degenerate implicit evolution inclusions, Electron. J. Differential Equations, 34 (2000), 20 pp.  Google Scholar

[10]

K. L. Kuttler and M. Shillor, Set-valued pseudomonotone maps and degenerate evolution inclusions, Commun. Contemp. Math., 1 (1999), 87-123.  doi: 10.1142/S0219199799000067.  Google Scholar

[11]

J.-L. Lions, Quelques Méthods de Résolution des Problèmes aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.  Google Scholar

[12]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Vol. 26, Advances in Mechanics and Mathematics, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.  Google Scholar

[13]

M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact, Vol. 655, Lecture Notes in Physics, Springer, Berlin, Heidelberg, 2004. doi: 10.1007/b99799.  Google Scholar

[14]

M.Sofonea, W. Han and M. Shillor, Analysis and Approximation of Contact Problems with Adhesion or Damage, Vol. 276, Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2006.  Google Scholar

[1]

Kevin T. Andrews, Kenneth L. Kuttler, Ji Li, Meir Shillor. Measurable solutions to general evolution inclusions. Evolution Equations & Control Theory, 2020, 9 (4) : 935-960. doi: 10.3934/eect.2020055
[2]

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

Noboru Okazawa, Kentarou Yoshii. Linear evolution equations with strongly measurable families and application to the Dirac equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 723-744. doi: 10.3934/dcdss.2011.4.723
[4]

Yong Zhou, V. Vijayakumar, R. Murugesu. Controllability for fractional evolution inclusions without compactness. Evolution Equations & Control Theory, 2015, 4 (4) : 507-524. doi: 10.3934/eect.2015.4.507
[5]

Leszek Gasiński, Nikolaos S. Papageorgiou. Periodic solutions for nonlinear nonmonotone evolution inclusions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 219-238. doi: 10.3934/dcdsb.2018015
[6]

Gabriel Ponce, Ali Tahzibi, Régis Varão. Minimal yet measurable foliations. Journal of Modern Dynamics, 2014, 8 (1) : 93-107. doi: 10.3934/jmd.2014.8.93
[7]

Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu. Periodic solutions for time-dependent subdifferential evolution inclusions. Evolution Equations & Control Theory, 2017, 6 (2) : 277-297. doi: 10.3934/eect.2017015
[8]

Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations & Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028
[9]

Tao Yu. Measurable sensitivity via Furstenberg families. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4543-4563. doi: 10.3934/dcds.2017194
[10]

Thomas Lorenz. Mutational inclusions: Differential inclusions in metric spaces. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 629-654. doi: 10.3934/dcdsb.2010.14.629
[11]

Marc Rauch. Variational principles for the topological pressure of measurable potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 367-394. doi: 10.3934/dcdss.2017018
[12]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098
[13]

Yajing Li, Yejuan Wang. The existence and exponential behavior of solutions to time fractional stochastic delay evolution inclusions with nonlinear multiplicative noise and fractional noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2665-2697. doi: 10.3934/dcdsb.2020027
[14]

Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020051
[15]

Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2503-2520. doi: 10.3934/jimo.2019066
[16]

Abdelkader Boucherif. Nonlocal problems for parabolic inclusions. Conference Publications, 2009, 2009 (Special) : 82-91. doi: 10.3934/proc.2009.2009.82
[17]

Mariusz Michta. On solutions to stochastic differential inclusions. Conference Publications, 2003, 2003 (Special) : 618-622. doi: 10.3934/proc.2003.2003.618
[18]

Michel H. Geoffroy, Alain Piétrus. A fast iterative scheme for variational inclusions. Conference Publications, 2009, 2009 (Special) : 250-258. doi: 10.3934/proc.2009.2009.250
[19]

Robert J. Kipka, Yuri S. Ledyaev. Optimal control of differential inclusions on manifolds. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4455-4475. doi: 10.3934/dcds.2015.35.4455
[20]

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

2019 Impact Factor: 0.953

Tools

Metrics

  • PDF downloads (58)
  • HTML views (251)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences