Electron. J. Diff. Equ., Vol. 2016 (2016), No. 90, pp. 1-12.

A general product measurability theorem with applications to variational inequalities

Kenneth L. Kuttler, Ji Li, Meir Shillor

This work establishes the existence of measurable weak solutions to evolution problems with randomness by proving and applying a novel theorem on product measurability of limits of sequences of functions. The measurability theorem is used to show that many important existence theorems within the abstract theory of evolution inclusions or equations have straightforward generalizations to settings that include random processes or coefficients. Moreover, the convex set where the solutions are sought is not fixed but may depend on the random variables. The importance of adding randomness lies in the fact that real world processes invariably involve randomness and variability. Thus, this work expands substantially the range of applications of models with variational inequalities and differential set-inclusions.

Submitted January 16, 2016. Published March 31, 2016.
Math Subject Classifications: 35R60, 60H15, 35R45, 35R70, 35S11.
Key Words: Partial differential inclusions; product measurability; variational inequalities; measurable selection.

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Kenneth L. Kuttler
Department of Mathematics
Brigham Young University
Provo, UT 84602, USA
email: klkuttle@math.byu.edu
Ji Li
Department of Mathematics
Michigan State University
East Lansing, MI 48824, USA
email: liji@math.msu.edu
Meir Shillor
Department of Mathematics and Statistics
Oakland University
Rochester, MI 48309, USA
email: shillor@oakland.edu

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