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.
 A. Energy estimates for bilinear form B.
 B. Existence of weak solutions for elliptic Dirichlet problem.
 C. Elliptic regularity.
 D. Maximum principles.
 E. Eigenfunctions of symmetric elliptic operator.
 F. Green formulas.
 15 Parabolic PDE.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Energy estimates for bilinear form B.

roposition

(Energy estimates for the bilinear form B). Let be the bilinear form given by the definition ( Bilinear form B ) and satisfy the definition ( Elliptic differential operator ). Then for any where the constants , , are dependent only on the functions and the set .

Remark

We may add to both sides and rewrite the second inequality as

Proof

Based on the definition ( Bilinear form B ) of we directly estimate Each of the integrals is dominated by for some constant dependent only on . Hence,

According to the definition ( Elliptic differential operator ) the matrix is uniformly positive definite for all . Hence, there exists a constant such that for any We integrate the above and use the definition ( Bilinear form B ) of :

We use the formula ( Cauchy inequality with epsilon ) to estimate the second term: and choose the so that We continue the estimation: where the constant depends only on the functions and the set .

 Notation. Index. Contents.