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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  December 2020 Early access  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

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