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.
 a. Finite differences in Sobolev spaces.
 b. Internal elliptic regularity.
 c. Boundary 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.

## Boundary elliptic regularity.

roposition

(Boundary elliptic regularity) Let be a bounded open set, , , and be a weak solution of the elliptic boundary problem (see the definition ( Elliptic differential operator )) Let be .

Then and where the constant depends only on and .

Remark

The above proposition is usually combined with the proposition ( Elliptic boundedness of inverse ) to get rid of the term.

Proof

The proof is a combination of proofs of the propositions ( Second order internal elliptic regularity ), ( High order internal elliptic regularity ) and ( Trace theorem ).

 Notation. Index. Contents.