Weak and strong formulations for stationary
variational inequality problem.

roblem

(Stationary variational
inequality problem) For a bounded set
with smooth boundary and given functions
and
find a function
satisfying the
relationships
where the operation
is given by the definition (
Bilinear form B
)
and the class of functions
is defined
by
We always assume that
.

We denote
the solution operator

Problem

(Strong formulation of
stationary problem) For a bounded set
with smooth boundary and given functions
and
find a function
satisfying the
relationships
in
and
where the operator
is defined by the formula (
Operator L
).

(Weak formulation of
stationary problem) For a bounded set
with smooth boundary and given functions
and
find a function
satisfying the
relationships
where the operation
is given by the definition (
Bilinear form B
)
and the class of functions
is defined
by