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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.

Existence of weak solutions for elliptic Dirichlet problem.


roposition

(Existence of weak solution for elliptic Dirichlet problem 1). Let $U$ be a bounded open subset of $\QTR{cal}{R}^{n}$ with a $C^{1}$ -boundary and MATH satisfy the definition ( Elliptic differential operator ). There exists a number $\gamma$ dependent only on $U$ and $L$ such that for any $\mu\geq\gamma$ and any function MATH there exists a weak solution of the problem MATH

Proof

The statement follows from the proposition ( Lax-Milgram theorem ) applied to the problem MATH The bilinear form is MATH We use the proposition ( Energy estimates for the bilinear form B ) and estimate MATH MATH MATH Hence, for MATH the form $B_{\mu}$ satisfies conditions of the proposition ( Lax-Milgram theorem ) in MATH . The mapping MATH is a bounded linear functional in MATH .

Definition

We introduce the following "adjoint" operator $L^{\ast}$ and a bilinear form $B^{\ast}$ : MATH

Definition

The function MATH is a weak solution of the problem MATH if MATH

Proposition

(Existence of weak solution for elliptic Dirichlet problem 2). Let $U$ be a bounded open subset of $\QTR{cal}{R}^{n}$ with a $C^{1}$ -boundary and MATH satisfy the definition ( Elliptic differential operator ).

1. Either

MATH (Elliptic alternative 1)
or
MATH (Elliptic alternative 2)

2. If the assertion ( Elliptic alternative 2 ) holds then MATH 3. The problem ( Elliptic Dirichlet problem ) has a weak solution if and only if MATH

Proof

According to the proposition ( Existence of weak solution for elliptic Dirichlet problem 1 ), there exists a mapping MATH where the $u$ is the weak solution of the problem MATH Hence, a function $w$ is a weak solution of the problem MATH if MATH or MATH The functions $f$ and $u$ are connected by $R_{\gamma}f=u$ iff MATH . According to the proof of the proposition ( Existence of weak solution for elliptic Dirichlet problem 1 ), the MATH satisfies conditions of the proposition ( Lax-Milgram theorem ): MATH for some constant $\theta>0$ . We estimate MATH According to the proposition ( Rellich-Kondrachov compactness theorem ) the last inequality implies that the operator $R_{\gamma}$ is compact. The rest follows from the proposition ( Fredholm alternative ).

Proposition

(Existence of weak solution for elliptic Dirichlet problem 3)

1. There exists at most countable set MATH such that the boundary value problem MATH has a unique weak solution for MATH iff MATH .

2. If MATH is infinite then MATH where the values $\lambda_{k}$ are nondecreasing and MATH .

Proof

We have established during the proof of the previous proposition that the operator MATH is compact in MATH if defined. The rest follows from the proposition ( Spectrum of compact operator ).

Proposition

(Elliptic boundedness of inverse) Let $u$ be a weak solution of MATH for MATH then MATH for a constant $C$ depending only on $\lambda,U$ and $L$ . The constant $C$ blows up if $\lambda$ approaches MATH .

Proof

The statement is a simple corollary of the proof of the proposition ( Existence of weak solution for elliptic Dirichlet problem 2 ).





Notation. Index. Contents.


















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