chapter  6
62 Pages

Differential inclusions of hemivariational inequality type

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


(u; v u) 0 for all v 2 X; (A)

where J


(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].