Lions–Lax–Milgram theorem

The lemma of Lax- Milgram, also set of Lax- Milgram, is a statement of functional analysis, a branch of mathematics, which is named after Peter Lax and Arthur Milgram. These two mathematicians proved in 1954 a first version of this lemma, which generalizes the statement of the theorem of Fréchet - Riesz on steady Sesquilinearformen. A more general version of the lemma was proved by Ivo Babuška, which is why this statement is also known as a set of Babuška -Lax - Milgram. Apply this statement in the theory of partial differential equations. With their help, existence and uniqueness statements about solutions are made by partial differential equations.

Formulation

Requirements

It is a Hilbert space and it is a sesquilinear form. In addition, applies one of the following equivalent conditions:

• Is continuous
• There is one with
• Is continuous for all and is continuous for all

Statement

If the above conditions are satisfied, then there exists a unique continuous linear operator, the equation

Satisfied for all. In addition: The norm of is limited by.

Special case: coercive sesquilinear

If the sesquilinear also coercive ( often referred to as strongly positive or elliptical), ie there are so

Holds, then is invertible.

Application to elliptic differential equations

To apply the Lemma of Lax- Milgram comes in the theory of partial differential equations. In particular, can be used for linear differential equations existence and uniqueness of a weak solution show if the above conditions are met. This will now be illustrated by the example of a uniformly elliptic second order differential equation.

Be

A uniformly elliptic differential operator of second order. That is, it applies to, and with it one exists, so that the main symbol for all the inequality

Met. With the help of the lemma of Lax- Milgram you can now show that the weak formulation of the Dirichlet problem

Exactly one solution in the Sobolev space and has. This means you look for all the test functions, the equation

Provides partial integration of the right side of the equation

If we now

We obtain a real-valued bilinear form, from which one can show the continuity with the help of Hölder 's inequality. The form is also coercive, which follows from the condition. Therefore, the bilinear form satisfies the conditions of the lemma of Lax- Milgram. It is now so looking for a solution of the equation

In which

Since the expression is linear and continuous, so is an element of the dual space, one can apply the representation theorem of Riesz and receives exactly one, such that for all. And due to the Lax- Milgram lemma, the equation

Exactly for all a solution.

Similarly, one can show the existence and uniqueness for Neumann boundary conditions.

Set of Babuška -Lax - Milgram

A generalization of the lemma of Lax- Milgram is the set of Babuška -Lax - Milgram. This was proved in 1971 by Ivo Babuška.

Let and be two Hilbert spaces and let a continuous bilinear form. Be also weakly coercive, ie, it exists, so that

And

Applies. Then there exists a unique continuous linear operator, the equation

To all and fulfills and for the operator norm, the inequality holds. In other words, there is exactly one solution for equations.