ABSTRACT
Chapter 6
Dierential inclusions of
hemivariational inequality type
The variational formulation of various boundary value problems in Mechan-
ics and Engineering governed by nonconvex, possibly nonsmooth energy
functionals (so-called superpotentials) leads to hemivariational inequalities
introduced by Panagiotopoulos; cf., e.g., [101, 181, 187, 188]. An abstract
formulation of a hemivariational inequality reads as follows:
Let X be a re exive Banach space and X
its dual, let A : X ! X
be some pseudomonotone and coercive operator (see section D) satisfying
certain continuity conditions, and let h 2 X
be some given element. Find
u 2 X such that
hAu h; v ui+ J
o
(u; v u) 0 for all v 2 X; (A)
where J
o
(u; v) denotes the generalized directional derivative in the sense of
Clarke (cf. [96]) of a locally Lipschitz functional J : X ! R: An equivalent
multivalued formulation of (A) is given by
Au+ @J(u) 3 h in X
; (B)
where @J(u) denotes Clarke's generalized gradient; cf. [96]. Abstract exis-
tence results for (A) (resp. (B)) can be found, e.g., in [181].
