Open main menu

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 itself and its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, thus a Banach space. Intuitively, a Sobolev space is a space of functions with 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.

Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that solutions of partial differential equations are naturally found in Sobolev spaces, rather than in spaces of continuous functions and with the derivatives understood in the classical sense.

Contents

MotivationEdit

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 C1 — 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 C1 (or C2, 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 L2-norm. It is therefore important to develop a tool for differentiating Lebesgue space functions.

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

 

where α a multi-index of order |α| = k and we are using the notation:

 

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

 

we call v the weak α-th partial derivative of u. If there exists a weak α-th partial derivative of u, 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 u ∈ Ck(Ω), then the classical and the weak derivative coincide. Thus, if v is a weak α-th partial derivative of u, we may denote it by Dαu := v.

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 p, see definition below).

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

Sobolev spaces with integer kEdit

One-dimensional caseEdit

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 of the function   is differentiable almost everywhere and is equal almost everywhere to the Lebesgue integral of its derivative (this gets rid of examples such as Cantor's function which are irrelevant to what the definition is trying to accomplish).

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

 

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 = 2Edit

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 examplesEdit

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. 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 caseEdit

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 functionsEdit

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]

ExamplesEdit

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 functionsEdit

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 boundaryEdit

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).

TracesEdit

Sobolev spaces are often considered when investigating partial differential equations. It is essential to consider boundary values of Sobolev functions. If u ∈ C(Ω), those boundary values are described by the restriction  . However, it is not clear how to describe values at the boundary for u ∈ Wk,p(Ω), 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 kEdit

Bessel potential spacesEdit

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.

  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 spacesEdit

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.[citation needed]((Check Example 9.1 in the Hitchhiker guide.))

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 operatorsEdit

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 = 2Edit

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 zeroEdit

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 embeddingsEdit

It is a natural question to ask if a Sobolev function is continuous or even continuously differentiable. Roughly speaking, sufficiently many weak derivatives or large p 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.

NotesEdit

  1. ^ Evans 1998, Chapter 5.2
  2. ^ a b c Adams 1975
  3. ^ Bergh & Löfström 1976
  4. ^ a b 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).

ReferencesEdit

External linksEdit