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 non-coercive stationary problem.

ondition

(Non-coercivity assumption 1) The coefficient of the formula ( Operator L ) satisfies for some constant .

Remark

(Preliminary reduction) It does not restrict generality to assume that for a constant .

Indeed, since is bounded there exists a constant such that then we set to be a solution of the problem and consider instead of .

Proposition

(Existence and uniqueness for non-coercive stationary problem.) Suppose that the condition ( Non-coercivity assumption 1 ) holds and the functions of the problem ( Stationary variational inequality problem ) satisfy Then there exists a unique solution of the problem ( Stationary variational inequality problem ) and

Proof

(Existence) Let is such that for some such that on . For example, given the condition ( Non-coercivity assumption 1 ), can be a large enough constant so that .

According to the proposition ( Energy estimates for the bilinear form B ) there exist constants such that

According to the proposition ( Existence and uniqueness for stationary problem ) there is a sequence such that

We now prove that a.e. in . Indeed, the are given by the relationships Note that we cannot apply the proposition ( Monotonicity of solution of stationary problem ) because the operators of the problems are different and because but . However, we act similarly to the proof of the proposition ( Monotonicity of solution of stationary problem ). We set in the first relationship and in the second: add: and move terms Note that . Hence Therefore, we deduce or where Hence, by the choice of in ,

Next, we prove that a.e. for Indeed, we use the recursion and assume that the same is proven for . The and are given by the relationships Hence, by the assumption of recursion, the proposition ( Monotonicity of solution of stationary problem ) applies.

Next, we show that for some constant using the recurrence argument.

We choose so that We set in the equation so that Note that hence We use : and apply the : By the recurrence assumption we have and by the choice of we have . Hence . Therefore,

We collect the previous results into the relationship

We now complete the proof of existence as follows. We fix in and rewrite it using Combining it with and boundedness of we deduce

Therefore, according to the proposition ( Weak compactness of bounded set ), then by the proposition ( Rellich-Kondrachov compactness theorem ), and boundedness of and We now pass the relationship : to the limit : thus

 Notation. Index. Contents.