# Sobolev space

In mathematics, a **Sobolev space** is a vector space of functions equipped with a norm that is a combination of *L ^{p}*-norms of the function itself as well as 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.

## Motivation

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 *C*^{1} — see Differentiability class). Differentiable functions are important in many areas, and in particular for differential equations. In the twentieth century, however, it was observed that the space *C*^{1} (or *C*^{2}, 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 *L*^{2}-norm. It is therefore important to develop a tool for differentiating Lebesgue space functions.

The integration by parts formula yields that for every *u* ∈ *C*^{k}(Ω), where *k* is a natural number and for all infinitely differentiable functions with compact support *φ* ∈ *C*_{c}^{∞}(Ω),

where *α* a multi-index of order |*α*| = *k* and Ω is an open subset in ℝ* ^{n}*. Here, the notation

is used.

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.
On the other hand, if *u* ∈ *C*^{k}(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 *D*^{α}*u* := *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 *W ^{k,p}*(Ω) combine the concepts of weak differentiability and Lebesgue norms.

## Sobolev spaces with integer Template:Mvar

### One-dimensional case

In the one-dimensional case (functions on **R**) the Sobolev space *W ^{ k,p}* is defined to be the subset of functions Template:Mvar in

*L*(

^{p}**R**) such that the function Template:Mvar and its weak derivatives up to some order Template:Mvar have a finite

*L*norm, for given

^{p}*p*(1 ≤

*p*≤ +∞). 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

*f*

^{ (k−1)}, the (

*k*− 1)-th derivative of the function Template:Mvar, 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 Template:!! ⋅ Template:!!_{k,p}, *W ^{ k,p}* 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 (see Normed vector space#Topological structure).

#### The case *p* = 2

Sobolev spaces with *p* = 2 (at leastTemplate:Clarify on a one-dimensional finite interval) 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:

*H*=^{ k}*W*^{ k,2}.

The space *H ^{ k}* can be defined naturally in terms of Fourier series whose coefficients decay sufficiently rapidly, namely,

where is the Fourier series of Template:Mvar. 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 *H ^{ k}* admits an inner product, like the space

*H*

^{ 0}=

*L*

^{2}. In fact, the

*H*inner product is defined in terms of the

^{ k}*L*

^{2}inner product:

The space *H ^{ k}* becomes a Hilbert space with this inner product.

#### Other examples

Some other Sobolev spaces permit a simpler description. For example, *W*^{ 1,1}(0, 1) is the space of absolutely continuous functions on (0, 1) (or rather, equivalence classes of functions that are equal almost everywhere to such), while *W*^{ 1,∞}(*I*) is the space of Lipschitz functions on Template:Mvar, for every interval Template:Mvar. All spaces *W*^{ k,∞} 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 *p* < +∞. (E.g., functions behaving like |*x*|^{−1/3} at the origin are in *L*^{2}, but the product of two such functions is not in *L*^{2}).

### Multidimensional case

The transition to multiple dimensions brings more difficulties, starting from the very definition. The requirement that *f*^{ (k−1)} be the integral of *f*^{ (k)} does not generalize, and the simplest solution is to consider derivatives in the sense of distribution theory.

A formal definition now follows. Let Ω be an open set in **R**^{n}, let Template:Mvar be a natural number and let 1 ≤ *p* ≤ +∞. The Sobolev space *W ^{ k,p}*(Ω) is defined to be the set of all functions Template:Mvar defined on Ω such that for every multi-index Template:Mvar with |

*α*| ≤

*k*, the mixed partial derivative

is both locally integrable and in *L ^{p}*(Ω), i.e.

That is, the Sobolev space *W ^{ k,p}*(Ω) is defined as

The natural number Template:Mvar is called the order of the Sobolev space *W ^{ k,p}*(Ω).

There are several choices for a norm for *W ^{ k,p}*(Ω). The following two are common and are equivalent in the sense of equivalence of norms:

and

With respect to either of these norms, *W ^{ k,p}*(Ω) is a Banach space. For

*p*< +∞,

*W*(Ω) is also a separable space. It is conventional to denote

^{ k,p}*W*

^{ k,2}(Ω) by

*H*(Ω) for it is a Hilbert space with the norm .

^{ k}^{[1]}

#### Approximation by smooth functions

Many of the properties of the Sobolev spaces cannot be seen directly from the definition. It is therefore interesting to investigate under which conditions a function *u* ∈ *W ^{ k,p}*(Ω) can be approximated by smooth functions. If Template:Mvar is finite and Ω is bounded with Lipschitz boundary, then for any

*u*∈

*W*(Ω) there exists an approximating sequence of functions

^{ k,p}*u*∈

_{m}*C*

^{∞}(Template:Overline), smooth up to the boundary such that:

^{[2]}

#### Examples

In higher dimensions, it is no longer true that, for example, *W*^{1,1} contains only continuous functions. For example, 1/|*x*| belongs to *W*^{1,1}(**B**^{3}) where **B**^{3} is the unit ball in three dimensions. For *k* > *n*/*p* the space *W ^{k,p}*(Ω) 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

*f*:

**B**

^{n}→

**R**∪ {+∞}, 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 Ω be an open set in **R**^{n} and 1 ≤ *p* ≤ +∞. If a function is in *W*^{ 1,p}(Ω), then, possibly after modifying the function on a set of measure zero, the restriction to almost every line parallel to the coordinate directions in **R**^{n} is absolutely continuous; what's more, the classical derivative along the lines that are parallel to the coordinate directions are in *L ^{p}*(Ω). Conversely, if the restriction of Template:Mvar to almost every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient ∇

*f*exists almost everywhere, and Template:Mvar is in

*W*

^{ 1,p}(Ω) provided Template:Mvar and |∇

*f*| are both in

*L*(Ω). In particular, in this case the weak partial derivatives of Template:Mvar and pointwise partial derivatives of Template:Mvar agree almost everywhere. The ACL characterization of the Sobolev spaces was established by Otto M. Nikodym (1933); see Template:Harv.

^{p}A stronger result holds in the case *p* > *n*. A function in *W*^{ 1,p}(Ω) is, after modifying on a set of measure zero, Hölder continuous of exponent *γ* = 1 − *n*/*p*, by Morrey's inequality. In particular, if *p* = +∞, then the function is Lipschitz continuous.

#### Functions vanishing at the boundary

Let Ω be an open set in **R**^{n}. The Sobolev space *W*^{ 1,2}(Ω) is also denoted by *H*^{1}(Ω). It is a Hilbert space, with an important subspace *H*Template:Su(Ω) defined to be the closure in *H*^{1}(Ω) of the infinitely differentiable functions compactly supported in Ω. The Sobolev norm defined above reduces here to

When Ω has a regular boundary, *H*Template:Su(Ω) can be described as the space of functions in *H*^{1}(Ω) that vanish at the boundary, in the sense of traces (see below). When *n* = 1, if Ω = (*a*, *b*) is a bounded interval, then *H*Template:Su(*a*, *b*) consists of continuous functions on [*a*, *b*] of the form

where the generalized derivative *f′* is in *L*^{2}(*a*, *b*) and has 0 integral, so that *f* (*b*) = *f* (*a*) = 0.

When Ω is bounded, the Poincaré inequality states that there is a constant *C* = *C*(Ω) such that

When Ω is bounded, the injection from *H*Template:Su(Ω) to *L*^{2}(Ω) 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 *L*^{2}(Ω) consisting of eigenvectors of the Laplace operator (with Dirichlet boundary condition).

## 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 *W ^{k,p}*(ℝ

*) can equivalently be defined as*

^{n}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 an open set Ω ⊆ ℝ* ^{n}*,

*H*(Ω) is the set of restrictions of functions from

^{s,p}*H*(ℝ

^{s,p}*) to Ω equipped with the norm*

^{n}Again, *H ^{s,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 *W ^{k,p}*(Ω) =

*H*(Ω) holds in the sense of equivalent norms, if Ω is domain with uniform

^{k,p}*C*

^{k}

*-boundary,*k

*a natural number and 1 < p < ∞. By the embeddings*

the Bessel potential spaces *H ^{s,p}*(ℝ

*) form a continuous scale between the Sobolev spaces*

^{n}*W*(ℝ

^{k,p}*). 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*

^{n}where:

### Sobolev–Slobodeckij spaces

Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder condition to the *L ^{p}*-setting.

^{[6]}For an open subset Ω of ℝ

*, 1 ≤ p < ∞, θ ∈ (0,1) and*

^{n}*f*∈

*L*(Ω), the

^{p}**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]} *W ^{s,p}*(Ω) is defined as

It is a Banach space for the norm

If the open subset Ω 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 *W*^{1,p}(Ω) is not even a vector subspace of *W ^{s,p}*(Ω) for 0 <

*s*< 1.

From an abstract point of view, the spaces *W ^{s,p}*(Ω) 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]}

## Traces

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* ∈ *W ^{k,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 *W*^{1,p}(Ω) for well-behaved Ω. Note that the trace operator *T* is in general not surjective, but for 1 < *p* < ∞ it maps onto the Sobolev-Slobodeckij space .

Intuitively, taking the trace costs 1/*p* of a derivative. The functions *u* in *W*^{1,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 *W*^{1,p}(Ω) can be approximated by smooth functions with compact support.

## Extension operators

If *X* is an open 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 *X* to functions of **R**^{n} such that:

We will call such an operator *A* an extension operator for *X*.

### Case of *p*=2

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

As in the section #Functions vanishing at the boundary, 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 X be uniformly C ^{m} regular, m ≥ s 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 s > ½. The map taking u to is continuous into if and only if s is not of the form n + ½ for n an integer.*

For a function *f* ∈ *L ^{p}*(Ω) on an open subset Ω of ℝ

*, its extension by zero*

^{n}is an element of *L ^{p}*(ℝ

*). Furthermore,*

^{n}In the case of the Sobolev space *W*^{1,p}(Ω) for 1 ≤ p ≤ ∞, extending a function *u* by zero will not necessarily yield an element of *W*^{1,p}(ℝ* ^{n}*). But if Ω is bounded with Lipschitz boundary (e.g. ∂Ω is C

*), 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*

^{1}^{[2]}

such that for each *u* ∈ *W*^{1,p}(Ω):
*Eu* = *u* 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 ℝ* ^{n}*.

## Sobolev embeddings

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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 *C*^{n,α} where *k* = *n* + α and 0 < α ≤ 1.) The Sobolev embedding theorem states that
if *k* ≥ *m* and *k* − *n*/*p* ≥ *m* − *n*/*q* then

and the embedding is continuous. Moreover if *k* > *m* and *k* − *n*/*p* > *m* −*n*/*q*
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* are continuous, so in particular this gives conditions on Sobolev spaces for various derivatives to be continuous. Informally these embeddings say that to convert an *L*^{p} 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 **R**^{n} Template:Harv.

## Notes

- ↑ Template:Harvnb
- ↑
^{2.0}^{2.1}^{2.2}Template:Harvnb - ↑ Template:Harvnb
- ↑
^{4.0}^{4.1}Template:Harvnb - ↑ 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).
- ↑ Template:Harvnb
- ↑ 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).

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