Measurable solutions for elliptic inclusions and quasistatic problems

Abstract

This work establishes the existence of variational solutions and their measurability to a very broad class of elliptic variational inequalities or set-inclusions under very general assumptions on the operators and the forcing functions and on their measurability. These results extend the current existence results for such problems and so the existence results are new, and so are the results when randomness is added. Moreover, it presents new measurability theorems when the operators and forcing functions are random processes. As an important application of the general existence result, it presents existence results for quasistatic problems in which the main operator is elliptic and time enters as a parameter or through dependence on auxiliary variables.