Sobolev space

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In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function.

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Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense.

Motivation

In this section and throughout the article is an open subset of

There are many criteria for smoothness of mathematical functions. The most basic criterion may be that of continuity. A stronger notion of smoothness is that of differentiability (because functions that are differentiable are also continuous) and a yet stronger notion of smoothness is that the derivative also be continuous (these functions are said to be of class see Differentiability classes). Differentiable functions are important in many areas, and in particular for differential equations. In the twentieth century, however, it was observed that the space (or , etc.) was not exactly the right space to study solutions of differential equations. The Sobolev spaces are the modern replacement for these spaces in which to look for solutions of partial differential equations.

Quantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms, rather than the uniform norm. A typical example is measuring the energy of a temperature or velocity distribution by an -norm. It is therefore important to develop a tool for differentiating Lebesgue space functions.

The integration by parts formula yields that for every , where is a natural number, and for all infinitely differentiable functions with compact support

where is a multi-index of order and we are using the notation:

The left-hand side of this equation still makes sense if we only assume to be locally integrable. If there exists a locally integrable function , such that

then we call the weak -th partial derivative of . If there exists a weak -th partial derivative of , then it is uniquely defined almost everywhere, and thus it is uniquely determined as an element of a Lebesgue space. On the other hand, if , then the classical and the weak derivative coincide. Thus, if is a weak -th partial derivative of , we may denote it by .

For example, the function

is not continuous at zero, and not differentiable at −1, 0, or 1. Yet the function

satisfies the definition for being the weak derivative of which then qualifies as being in the Sobolev space (for any allowed , see definition below).

The Sobolev spaces combine the concepts of weak differentiability and Lebesgue norms.

Sobolev spaces with integer k

One-dimensional case

In the one-dimensional case the Sobolev space for is defined as the subset of functions in such that and its weak derivatives up to order have a finite Lp norm. As mentioned above, some care must be taken to define derivatives in the proper sense. In the one-dimensional problem it is enough to assume that the -th derivative is differentiable almost everywhere and is equal almost everywhere to the Lebesgue integral of its derivative (this excludes irrelevant examples such as Cantor's function).

With this definition, the Sobolev spaces admit a natural norm,

One can extend this to the case , with the norm then defined using the essential supremum by

Equipped with the norm becomes a Banach space. It turns out that it is enough to take only the first and last in the sequence, i.e., the norm defined by

is equivalent to the norm above (i.e. the induced topologies of the norms are the same).

The case p = 2

Sobolev spaces with p = 2 are especially important because of their connection with Fourier series and because they form a Hilbert space. A special notation has arisen to cover this case, since the space is a Hilbert space:

The space can be defined naturally in terms of Fourier series whose coefficients decay sufficiently rapidly, namely,

where is the Fourier series of and denotes the 1-torus. As above, one can use the equivalent norm

Both representations follow easily from Parseval's theorem and the fact that differentiation is equivalent to multiplying the Fourier coefficient by in.

Furthermore, the space admits an inner product, like the space In fact, the inner product is defined in terms of the inner product:

The space becomes a Hilbert space with this inner product.

Other examples

In one dimension, some other Sobolev spaces permit a simpler description. For example, is the space of absolutely continuous functions on (0, 1) (or rather, equivalence classes of functions that are equal almost everywhere to such), while is the space of Lipschitz functions on I, for every interval I. However, these properties are lost or not as simple for functions of more than one variable.

All spaces are (normed) algebras, i.e. the product of two elements is once again a function of this Sobolev space, which is not the case for (E.g., functions behaving like |x|−1/3 at the origin are in but the product of two such functions is not in ).

Multidimensional case

The transition to multiple dimensions brings more difficulties, starting from the very definition. The requirement that be the integral of does not generalize, and the simplest solution is to consider derivatives in the sense of distribution theory.

A formal definition now follows. Let The Sobolev space is defined to be the set of all functions on such that for every multi-index with the mixed partial derivative

exists in the weak sense and is in i.e.

That is, the Sobolev space is defined as

The natural number is called the order of the Sobolev space

There are several choices for a norm for The following two are common and are equivalent in the sense of equivalence of norms:

and

With respect to either of these norms, is a Banach space. For is also a separable space. It is conventional to denote by for it is a Hilbert space with the norm . [1]

Approximation by smooth functions

It is rather hard to work with Sobolev spaces relying only on their definition. It is therefore interesting to know that by theorem of Meyers and Serrin a function can be approximated by smooth functions. This fact often allows us to translate properties of smooth functions to Sobolev functions. If is finite and is open, then there exists for any an approximating sequence of functions such that:

If has Lipschitz boundary, we may even assume that the are the restriction of smooth functions with compact support on all of [2]

Examples

In higher dimensions, it is no longer true that, for example, contains only continuous functions. For example, where is the unit ball in three dimensions. For k > n/p the space will contain only continuous functions, but for which k this is already true depends both on p and on the dimension. For example, as can be easily checked using spherical polar coordinates for the function defined on the n-dimensional ball we have:

Intuitively, the blow-up of f at 0 "counts for less" when n is large since the unit ball has "more outside and less inside" in higher dimensions.

Absolutely continuous on lines (ACL) characterization of Sobolev functions

Let If a function is in then, possibly after modifying the function on a set of measure zero, the restriction to almost every line parallel to the coordinate directions in is absolutely continuous; what's more, the classical derivative along the lines that are parallel to the coordinate directions are in Conversely, if the restriction of to almost every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient exists almost everywhere, and is in provided In particular, in this case the weak partial derivatives of and pointwise partial derivatives of agree almost everywhere. The ACL characterization of the Sobolev spaces was established by Otto M. Nikodym (1933); see ( Maz'ya 1985 , §1.1.3).

A stronger result holds when A function in is, after modifying on a set of measure zero, Hölder continuous of exponent by Morrey's inequality. In particular, if then the function is Lipschitz continuous.

Functions vanishing at the boundary

The Sobolev space is also denoted by It is a Hilbert space, with an important subspace defined to be the closure of the infinitely differentiable functions compactly supported in in The Sobolev norm defined above reduces here to

When has a regular boundary, can be described as the space of functions in that vanish at the boundary, in the sense of traces (see below). When if is a bounded interval, then consists of continuous functions on of the form

where the generalized derivative is in and has 0 integral, so that

When is bounded, the Poincaré inequality states that there is a constant such that:

When is bounded, the injection from to is compact. This fact plays a role in the study of the Dirichlet problem, and in the fact that there exists an orthonormal basis of consisting of eigenvectors of the Laplace operator (with Dirichlet boundary condition).

Traces

Sobolev spaces are often considered when investigating partial differential equations. It is essential to consider boundary values of Sobolev functions. If , those boundary values are described by the restriction . However, it is not clear how to describe values at the boundary for , as the n-dimensional measure of the boundary is zero. The following theorem [2] resolves the problem:

Trace Theorem. Assume Ω is bounded with Lipschitz boundary. Then there exists a bounded linear operator such that

Tu is called the trace of u. Roughly speaking, this theorem extends the restriction operator to the Sobolev space for well-behaved Ω. Note that the trace operator T is in general not surjective, but for 1 < p < ∞ it maps continuously onto the Sobolev-Slobodeckij space

Intuitively, taking the trace costs 1/p of a derivative. The functions u in W1,p(Ω) with zero trace, i.e. Tu = 0, can be characterized by the equality

where

In other words, for Ω bounded with Lipschitz boundary, trace-zero functions in can be approximated by smooth functions with compact support.

Sobolev spaces with non-integer k

Bessel potential spaces

For a natural number k and 1 < p < ∞ one can show (by using Fourier multipliers [3] [4] ) that the space can equivalently be defined as

with the norm

This motivates Sobolev spaces with non-integer order since in the above definition we can replace k by any real number s. The resulting spaces

are called Bessel potential spaces [5] (named after Friedrich Bessel). They are Banach spaces in general and Hilbert spaces in the special case p = 2.

For is the set of restrictions of functions from to Ω equipped with the norm

.

Again, Hs,p(Ω) is a Banach space and in the case p = 2 a Hilbert space.

Using extension theorems for Sobolev spaces, it can be shown that also Wk,p(Ω) = Hk,p(Ω) holds in the sense of equivalent norms, if Ω is domain with uniform Ck-boundary, k a natural number and 1 < p < ∞. By the embeddings

the Bessel potential spaces form a continuous scale between the Sobolev spaces From an abstract point of view, the Bessel potential spaces occur as complex interpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms it holds that

where:

Sobolev–Slobodeckij spaces

Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder condition to the Lp-setting. [6] For and the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by

Let s > 0 be not an integer and set . Using the same idea as for the Hölder spaces, the Sobolev–Slobodeckij space [7] is defined as

It is a Banach space for the norm

If is suitably regular in the sense that there exist certain extension operators, then also the Sobolev–Slobodeckij spaces form a scale of Banach spaces, i.e. one has the continuous injections or embeddings

There are examples of irregular Ω such that is not even a vector subspace of for 0 < s < 1 (see Example 9.1 of [8] )

From an abstract point of view, the spaces coincide with the real interpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms the following holds:

.

Sobolev–Slobodeckij spaces play an important role in the study of traces of Sobolev functions. They are special cases of Besov spaces. [4]

Extension operators

If is a domain whose boundary is not too poorly behaved (e.g., if its boundary is a manifold, or satisfies the more permissive "cone condition") then there is an operator A mapping functions of to functions of such that:

  1. Au(x) = u(x) for almost every x in and
  2. is continuous for any 1 ≤ p ≤ ∞ and integer k.

We will call such an operator A an extension operator for

Case of p = 2

Extension operators are the most natural way to define for non-integer s (we cannot work directly on since taking Fourier transform is a global operation). We define by saying that if and only if Equivalently, complex interpolation yields the same spaces so long as has an extension operator. If does not have an extension operator, complex interpolation is the only way to obtain the spaces.

As a result, the interpolation inequality still holds.

Extension by zero

Like above, we define to be the closure in of the space of infinitely differentiable compactly supported functions. Given the definition of a trace, above, we may state the following

Theorem. Let be uniformly Cm regular, ms and let P be the linear map sending u in to
where d/dn is the derivative normal to G, and k is the largest integer less than s. Then is precisely the kernel of P.

If we may define its extension by zero in the natural way, namely

Theorem. Let The map is continuous into if and only if s is not of the form for n an integer.

For f  Lp(Ω) its extension by zero,

is an element of Furthermore,

In the case of the Sobolev space W1,p(Ω) for 1  p  ∞, extending a function u by zero will not necessarily yield an element of But if Ω is bounded with Lipschitz boundary (e.g. ∂Ω is C1), then for any bounded open set O such that Ω⊂⊂O (i.e. Ω is compactly contained in O), there exists a bounded linear operator [2]

such that for each a.e. on Ω, Eu has compact support within O, and there exists a constant C depending only on p, Ω, O and the dimension n, such that

We call Eu an extension of u to

Sobolev embeddings

It is a natural question to ask if a Sobolev function is continuous or even continuously differentiable. Roughly speaking, sufficiently many weak derivatives (i.e. large k) result in a classical derivative. This idea is generalized and made precise in the Sobolev embedding theorem.

Write for the Sobolev space of some compact Riemannian manifold of dimension n. Here k can be any real number, and 1  p  ∞. (For p = ∞ the Sobolev space is defined to be the Hölder space Cn where k = n + α and 0 < α  1.) The Sobolev embedding theorem states that if and then

and the embedding is continuous. Moreover, if and then the embedding is completely continuous (this is sometimes called Kondrachov's theorem or the Rellich-Kondrachov theorem). Functions in have all derivatives of order less than m continuous, so in particular this gives conditions on Sobolev spaces for various derivatives to be continuous. Informally these embeddings say that to convert an Lp estimate to a boundedness estimate costs 1/p derivatives per dimension.

There are similar variations of the embedding theorem for non-compact manifolds such as ( Stein 1970 ). Sobolev embeddings on that are not compact often have a related, but weaker, property of cocompactness.

Notes

  1. Evans 1998 , Chapter 5.2
  2. 1 2 3 Adams 1975
  3. Bergh & Löfström 1976
  4. 1 2 Triebel 1995
  5. Bessel potential spaces with variable integrability have been independently introduced by Almeida & Samko (A. Almeida and S. Samko, "Characterization of Riesz and Bessel potentials on variable Lebesgue spaces", J. Function Spaces Appl. 4 (2006), no. 2, 113–144) and Gurka, Harjulehto & Nekvinda (P. Gurka, P. Harjulehto and A. Nekvinda: "Bessel potential spaces with variable exponent", Math. Inequal. Appl. 10 (2007), no. 3, 661–676).
  6. Lunardi 1995
  7. In the literature, fractional Sobolev-type spaces are also called Aronszajn spaces, Gagliardo spaces or Slobodeckij spaces, after the names of the mathematicians who introduced them in the 1950s: N. Aronszajn ("Boundary values of functions with finite Dirichlet integral", Techn. Report of Univ. of Kansas 14 (1955), 77–94), E. Gagliardo ("Proprietà di alcune classi di funzioni in più variabili", Ricerche Mat. 7 (1958), 102–137), and L. N. Slobodeckij ("Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations", Leningrad. Gos. Ped. Inst. Učep. Zap. 197 (1958), 54–112).
  8. "Hitchhikerʼs guide to the fractional Sobolev spaces". Bulletin des Sciences Mathématiques. 136 (5): 521–573. 2012-07-01. doi:10.1016/j.bulsci.2011.12.004. ISSN   0007-4497.

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