Sobolev space

A Sobolev space, also Sobolev space ( according to Sergei Sobolev Lvovitch, as transliteration and transcription in English Sobolev ), in mathematics, a function space of weakly differentiable functions, which is also a Banach space. The concept was essentially driven by the systematic theory of variational calculus in the early 20th century. This minimizes functionals functions. Today Sobolev spaces form the basis of the solution theory of partial differential equations.

• 2.1 trace operator
• 2.2 Sobolev space with zero boundary conditions

• 3.1 Sobolev number
• 3.2 Sobolev embedding theorem of
• 3.3 embedding theorem of Rellich

• 4.1 Definition
• 4.2 Dual and Hilbert space

Sobolev spaces of integer order

There were open and.

As a Sobolev space we denote the space of those real-valued functions whose partial mixed weak derivatives up to order are in the Lebesgue space.

For the notation is also common.

Sobolev norm

For functions we define the norm by

This is a multi- index with and. Continues.

The Sobolev norm is therefore just the sum of the norms of all possible combinations of partial derivatives up to the nth order. The Sobolev space or with respect to the respective Sobolev norm complete.

Definition as a topological degree

Let us now consider the space of functions whose partial derivatives are to grade in, and denote this function space with. As different functions are never mutually equivalent ( see also Lp- space ), you can embed it in, and it holds the following inclusion

The room is not complete wrt the standard. Rather, its completion is straight. The partial derivatives up to order k can be continued as continuous operators on these Sobolev space of continuous clear. These sequels are just the weak derivatives.

Thus we obtain an alternative definition of Sobolev spaces. By the theorem of Meyers - Serrin it is equivalent to the above definition.

Properties

As already mentioned, with the norm a complete vector space, so that is a Banach space. For he is even reflexive.

For the standard is by the scalar product

Induced. is therefore a Hilbert space, and you also writes.

Boundary value problems

The weak derivative or the Sobolev spaces have been developed for solving partial differential equations. However, there is in solving boundary value problems still a difficulty. The weakly differentiable functions are just like the functions not defined on null sets. The expression for and thus yields only once no sense. For this problem, the restriction map has been generalized to the track operator.

Trace operator

Be a bounded domain with boundary. Then there exists a bounded, linear operator

So that

And

Applies to all. The constant depends only on and off. The operator is called trace operator.

Sobolev space with zero boundary conditions

With is called the conclusion of the test function space in. This means if and only if there exists a sequence with in

For one can prove that this quantity are exactly the Sobolev functions with zero boundary conditions. Thus if one edge, then if and only if the following holds in the sense of traces.

Embedding theorems

Sobolev number

Each Sobolev space with assigns you to a number that is important in the context of embedding sentences. Man sets

And calls this number the Sobolev number.

Embedding theorem of Sobolev

There are several inter-related statements, which we denote by embedding theorem of Sobolev embedding theorem sobolevscher or with Sobolev 's lemma. Be an open and bounded subset of, and the Sobolev number to. For exists a continuous embedding

Are equipped with or with the supremum norm. In other words, each equivalence class has a representative in. Applies contrast so you can at least continuously embedded in the space for all, being set.

From the Sobolev embedding theorem we conclude that there is a continuous embedding

Available for all.

Embedding theorem of Rellich

Be open and bounded and. Then the embedding

A linear compact operator. It denotes the identity map.

Sobolev space of real-valued order

Definition

Often Sobolev spaces are used with real exponents. These are defined in the whole space case, the Fourier transform of the function involved. The Fourier transform is designated herein. For a function is an item of, if

Applies. Due to the identity of these are for the same rooms, which were defined in the first section. with

Is a Hilbert space. The norm is given by

For a smooth bordered detes, bounded domain of the space is defined as the set of all that can be continued to a ( on defined ) function.

For one can also define Sobolev spaces. However, this purpose is necessary to resort to the theory of distributions. Be the space of tempered distributions, then for all by

Defined.

Dual and Hilbert space

If we consider the Banach space with the scalar product then its dual space. However, to keep the rooms with the help of the scalar product

Understood as a Hilbert space. Since Hilbert spaces are dual to itself, and to ( with respect to different products ) dual now. However, one can, and with the aid of isomorphism

Identify. In this way, the spaces and the isomorphism

Identify.