Sobolev space

Encyclopedia

In mathematics

, a

of functions equipped with a norm that is a combination of

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 equation

s, 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 function

s and with the derivative

s understood in the classical sense.

. 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

). Differentiable functions are important in many areas, and in particular for differential equation

s. On the other hand, 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

The integration by parts

formula yields that for every

and for all infinitely differentiable functions with compact support

,

where

is used.

The left-hand side of this equation still makes sense if we only assume

we call

of

.

On the other hand, if

The Sobolev spaces

belongs to

Here, Ω is an open set in ℝ

There are several choices for a norm for

and

With respect to either of these norms,

with the norm .

) that the space

with the norm

.

This motivates Sobolev spaces with non-integer order since in the above definition we can replace

are called Bessel potential spaces (named after Friedrich Bessel

) and are denoted by

For an open set Ω ⊆ ℝ

.

Again,

Using extension theorems for Sobolev spaces, it can be shown that also

s

the Bessel potential spaces

to the

.

Let be not an integer and set . Using the same idea as for the Hölder spaces, the

.

It is a Banach space for the norm

.

The Sobolev–Slobodeckij spaces give a second continuous scale between the Sobolev spaces, i.e. one has the embedding

s

.

From an abstract point of view, the spaces

.

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

Intuitively, taking the trace costs

The functions

where

In other words, for Ω bounded with Lipschitz boundary, trace-zero functions in

is an element of

In the case of the Sobolev space

such that

.

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a

**Sobolev space**is a vector spaceVector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

of functions equipped with a norm that is a combination of

*L*-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense^{p}Weak derivative

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space L^1. See distributions for an even more general definition.- Definition :Let u be a function in the...

to make the space complete, thus a Banach space

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

. Intuitively, a Sobolev space is a space of functions with sufficiently many derivatives for some application domain, such as partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s, and equipped with a norm that measures both the size and regularity of a function.

Sobolev spaces are named after the Russian mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

Sergei Sobolev

Sergei Lvovich Sobolev

Sergei Lvovich Sobolev was a Soviet mathematician working in mathematical analysis and partial differential equations. He was born in St. Petersburg, and died in Moscow.-Work:...

. Their importance comes from the fact that solutions of partial differential equations are naturally found in Sobolev spaces, rather than in spaces of continuous function

Continuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

s and with the derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

s understood in the classical sense.

## Motivation

There are many criteria for smoothness of mathematical functions. The most basic criterion may be that of continuityContinuous function

In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

. 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 smooth functionSmooth function

In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...

). Differentiable functions are important in many areas, and in particular for differential equation

Differential equation

A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...

s. On the other hand, 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 functions.The integration by parts

Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...

formula yields that for every

*u*∈*C*^{k}(Ω), where*k*is a natural numberNatural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

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*. Here, the notation*^{n}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 thatwe call

*v*the weak*α*-th partial derivativeWeak derivative

In mathematics, a weak derivative is a generalization of the concept of the derivative of a function for functions not assumed differentiable, but only integrable, i.e. to lie in the Lebesgue space L^1. See distributions for an even more general definition.- Definition :Let u be a function in the...

of

*u*. If there exists a weak*α*-th partial derivative of*u*, then it is uniquely defined almost everywhereAlmost everywhere

In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...

.

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*.The Sobolev spaces

*W*^{k,p}(Ω) combine the concepts of weak differentiability and Lebesgue norms.### Definition

The Sobolev space*W*^{k,p}(Ω) is defined to be the set of all functions*u*∈*L*^{p}(Ω) such that for every multi-index*α*with |*α*| ≤*k*, the weak partial derivativePartial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

belongs to

*L*^{p}(Ω), i.e.Here, Ω is an open set in ℝ

*and 1 ≤*^{n}*p*≤ +∞. The natural numberNatural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

*k*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 finite*p*,*W*^{k,p}(Ω) is also a separable space. It is conventional to denote*W*^{k,2}(Ω) by*H*^{k}(Ω) for it is a Hilbert spaceHilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

with the norm .

### Approximation by smooth functions

A lot of 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*p*is finite and Ω is bounded with Lipschitz boundary, then for any*u*∈*W*^{k,p}(Ω) there exists an approximating sequence of functions*u*_{m}∈*C*^{∞}, smooth up to the boundary such that ||*u*_{m}-*u*||_{}→ 0.### Bessel potential spaces

For a natural number*k*and one can show (by using Fourier multipliersMultiplier (Fourier analysis)

In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions. These operators act on a function by altering its Fourier transform. Specifically they multiply the Fourier transform of a function by a specified function known as the multiplier or symbol...

) 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 spacesare called Bessel potential spaces (named after Friedrich Bessel

Friedrich Bessel

-References:* John Frederick William Herschel, A brief notice of the life, researches, and discoveries of Friedrich Wilhelm Bessel, London: Barclay, 1847 -External links:...

) and are denoted by

*H*^{s,p}(ℝ*). They are Banach spaces in general and Hilbert spaces in the special case*^{n}*p = 2*.For an open set Ω ⊆ ℝ

*,*^{n}*H*^{s,p}(Ω) is the set of restrictions of functions from*H*^{s,p}(ℝ*) 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*^{k,p}(Ω) holds in the sense of equivalent norms, if Ω is domain with uniform*C*^{k}-boundary,*k*a natural number and . By the embeddingEmbedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

s

the Bessel potential spaces

*H*^{s,p}(ℝ*) form a continuous scale between the Sobolev spaces*^{n}*W*^{k,p}(ℝ*). 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}### Sobolev–Slobodeckij spaces

Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder conditionHölder condition

In mathematics, a real or complex-valued function ƒ on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, \alpha , such that...

to the

*L*p-setting. For an open subset Ω of ℝ^{}*, , θ ∈ (0,1) and*^{n}*f*∈*L*^{p}(Ω), the**Slobodeckij seminorm**(roughly analogous to the Hölder seminorm) is defined by.

Let be not an integer and set . Using the same idea as for the Hölder spaces, the

**Sobolev–Slobodeckij space***W*^{s, p}(Ω) is defined as.

It is a Banach space for the norm

.

The Sobolev–Slobodeckij spaces give a second continuous scale between the Sobolev spaces, i.e. one has the embedding

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

s

.

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.

## 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 resolves the problem:-
**Trace Theorem.**Assume Ω is bounded with Lipschitz boundary. Then there exists a bounded linear operator such that

- and

*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 operatorTrace operator

In mathematics, the concept of trace operator plays an important role in studying the existence and uniqueness of solutions to boundary value problems, that is, to partial differential equations with prescribed boundary conditions...

*T*is in general not surjective, but maps for*p*∈ (1,∞) 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 equalitywhere

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.## Extensions

For a function*f*∈*L*^{p}(Ω) on an open set Ω ∈ 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), extending a function*u*by zero will not necessarily yield an element of*W*^{1,p}(ℝ*). But for Ω bounded with Lipschitz boundary, there exists for every 1 ≤ p ≤ ∞ a bounded extension operator*^{n}such that

*Eu*=*u*on Ω,*Eu*has compact support and- there exists a constant
*c*depending only on Ω and the dimension*n*, such that

## 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 or large*p*result in a classical derivative. This idea is generalized and made precise in the Sobolev embedding theoremSobolev inequality

In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under...

.