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 be a bounded open subset of with a -boundary and satisfy the definition ( Elliptic differential operator ). There exists a number dependent only on and such that for any and any function there exists a weak solution of the problem

Proof

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

Definition

We introduce the following "adjoint" operator and a bilinear form :

Definition

The function is a weak solution of the problem if

Proposition

(Existence of weak solution for elliptic Dirichlet problem 2). Let be a bounded open subset of with a -boundary and satisfy the definition ( Elliptic differential operator ).

1. Either

 (Elliptic alternative 1)
or
 (Elliptic alternative 2)

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

Proof

According to the proposition ( Existence of weak solution for elliptic Dirichlet problem 1 ), there exists a mapping where the is the weak solution of the problem Hence, a function is a weak solution of the problem if or The functions and are connected by iff . According to the proof of the proposition ( Existence of weak solution for elliptic Dirichlet problem 1 ), the satisfies conditions of the proposition ( Lax-Milgram theorem ): for some constant . We estimate According to the proposition ( Rellich-Kondrachov compactness theorem ) the last inequality implies that the operator 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 such that the boundary value problem has a unique weak solution for iff .

2. If is infinite then where the values are nondecreasing and .

Proof

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

Proposition

(Elliptic boundedness of inverse) Let be a weak solution of for then for a constant depending only on and . The constant blows up if approaches .

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.