I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 1 Calculational Linear Algebra.
 2 Wavelet Analysis.
 3 Finite element method.
 4 Construction of approximation spaces.
 5 Time discretization.
 6 Variational inequalities.
 A. Stationary variational inequalities.
 a. Weak and strong formulations for stationary variational inequality problem.
 b. Existence and uniqueness for coercive stationary problem.
 c. Penalized stationary problem.
 d. Proof of existence for stationary problem.
 e. Estimate of penalization error for stationary problem.
 f. Monotonicity of solution of stationary problem.
 g. Existence and uniqueness for non-coercive stationary problem.
 B. Evolutionary variational inequalities.
 VIII. Bibliography
 Notation. Index. Contents.

## 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 ).

Proposition

For a function the formulations ( Strong formulation of stationary problem ) and ( Stationary variational inequality problem ) are equivalent.

Proof

See the remarks in the section ( Variational inequalities ) and the section ( Elliptic PDE ).

Problem

(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 Proposition

Assume that Then the formulations ( Stationary variational inequality problem ) and ( Weak formulation of stationary problem ) are equivalent.

Proof

If and then and follows. The prove in the opposite direction is the same.

 Notation. Index. Contents.