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.

## Existence and uniqueness for coercive stationary problem.

he following condition complements the second conclusion of the proposition ( Energy estimates for the bilinear form B ).

Condition

(Assumption of coercivity 1) There is a number such that

Proposition

(Existence and uniqueness for stationary problem) If the coefficients of satisfy the definition ( Elliptic differential operator ) and the condition ( Assumption of coercivity 1 ) then the problem ( Stationary variational inequality problem ) has a unique solution.

Proof

We prove uniqueness. The proof of existence is postponed until the section ( Proof of existence for stationary problem ).

Assume that there are two solutions and : We set in and in and add the results Hence, by the condition ( Assumption of coercivity 1 ) Thus

 Notation. Index. Contents.