Measurable solutions for elliptic and evolution inclusions

Kenneth Kuttler

doi:10.3934/eect.2020041

Article Contents

2020, Volume 9, Issue 4: 1041-1055. Doi: 10.3934/eect.2020041

This issue Previous Article Relaxation of optimal control problems driven by nonlinear evolution equations Next Article Differential inclusion problems with convolution and discontinuous nonlinearities

Measurable solutions for elliptic and evolution inclusions

Kenneth Kuttler^,

249 Havenside Court, Grantsville, Utah 84029, USA

^* Kenneth Kuttler
^* 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

Early access: March 2020

Published: December 2020

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Abstract

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.

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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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[9]	K. Kuttler, Non-degenerate implicit evolution inclusions, Electron. J. Differential Equations, 34 (2000), 20 pp.
[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.
[11]	J.-L. Lions, Quelques Méthods de Résolution des Problèmes aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.
[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.
[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.
[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.