This book focuses on a large class of multi-valued variational
differential inequalities and inclusions of stationary and evolutionary
types with constraints reflected by subdifferentials of convex
functionals. Its main goal is to provide a systematic, unified, and
relatively self-contained exposition of existence, comparison and
enclosure principles, together with other qualitative properties of
multi-valued variational inequalities and inclusions. The problems under
consideration are studied in different function spaces such as Sobolev
spaces, Orlicz-Sobolev spaces, Sobolev spaces with variable exponents,
and Beppo-Levi spaces.
A general and comprehensive sub-supersolution method (lattice method) is
developed for both stationary and evolutionary multi-valued variational
inequalities, which preserves the characteristic features of the
commonly known sub-supersolution method for single-valued, quasilinear
elliptic and parabolic problems. This method provides a powerful tool
for studying existence and enclosure properties of solutions when the
coercivity of the problems under consideration fails. It can also be
used to investigate qualitative properties such as the multiplicity and
location of solutions or the existence of extremal solutions.
This is the first in-depth treatise on the sub-supersolution (lattice)
method for multi-valued variational inequalities without any variational
structures, together with related topics. The choice of the included
materials and their organization in the book also makes it useful and
accessible to a large audience consisting of graduate students and
researchers in various areas of Mathematical Analysis and Theoretical
Physics.
