Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 1 Real Variable.
 2 Laws of large numbers.
 3 Characteristic function.
 4 Central limit theorem (CLT) II.
 5 Random walk.
 6 Conditional probability II.
 7 Martingales and stopping times.
 8 Markov process.
 9 Levy process.
 10 Weak derivative. Fundamental solution. Calculus of distributions.
 11 Functional Analysis.
 12 Fourier analysis.
 13 Sobolev spaces.
 14 Elliptic PDE.
 15 Parabolic PDE.
 A. Galerkin approximation for parabolic Dirichlet problem.
 B. Energy estimates for Galerkin approximate solution.
 C. Existence of weak solution for parabolic Dirichlet problem.
 D. Parabolic regularity.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Energy estimates for Galerkin approximate solution.

roposition

(Energy estimates for the Galerkin approximate solution). The solution of the equations ( Galerkin problem ) satisfies the estimates for all and constants depending only on and the functions .

Proof

We multiply the equations ( Galerkin problem ) by and sum for . We obtain We add the relationship to the second estimate of the proposition ( Energy estimates for the bilinear form B ): and obtain We combine the above with (*) and obtain We apply the formula ( Cauchy inequality ) to the term :

We drop the term in the formula (**), recall the starting condition for at : and apply the proposition ( Differential inequality 2 ): Thus, we proved the first desired inequality.

Next, we drop the term from the formula (**) and integrate in : hence, we proved the second inequality.

To prove the final inequality, we seek to estimate the quantity

for any such that See the remark in the section ( Dual Sobolev spaces ).

We introduce : the projection on the linear span of in . We deduce from the formula ( Galerkin problem ) that and apply the formula ( Holder inequality ) and the proposition ( Energy estimates for the bilinear form B ) to estimate the RHS: Note that , hence We integrate the last inequality in and use the first inequality of this proposition

 Notation. Index. Contents.