Sobolev space
Encyclopedia
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 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 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 C1 — see smooth function
). 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 L2-norm. It is therefore important to develop a tool for differentiating Lebesgue functions.
The integration by parts
formula yields that for every u ∈ Ck(Ω), where k is a natural number
and for all infinitely differentiable functions with compact support φ ∈ Cc∞(Ω),
,
where α a multi-index of order |α| = k and Ω is an open subset in ℝn. 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 ∈ 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.
The Sobolev spaces Wk,p(Ω) combine the concepts of weak differentiability and Lebesgue norms.
belongs to Lp(Ω), i.e.
Here, Ω is an open set in ℝn and 1 ≤ p ≤ +∞. The natural number
k is called the order of the Sobolev space Wk,p(Ω).
There are several choices for a norm for Wk,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, Wk,p(Ω) is a Banach space. For finite p, Wk,p(Ω) is also a separable space. It is conventional to denote Wk,2(Ω) by Hk(Ω) for it is a Hilbert space
with the norm
.
→ 0.
) that the space Wk,p(ℝn) 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 (named after Friedrich Bessel
) and are denoted by Hs,p(ℝn). They are Banach spaces in general and Hilbert spaces in the special case p = 2 .
For an open set Ω ⊆ ℝn, Hs,p(Ω) is the set of restrictions of functions from Hs,p(ℝn) 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 . By the embedding
s
the Bessel potential spaces Hs,p(ℝn) form a continuous scale between the Sobolev spaces Wk,p(ℝn). 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
to the Lp-setting. For an open subset Ω of ℝn, , θ ∈ (0,1) and f ∈ Lp(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 Ws, 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
s
.
From an abstract point of view, the spaces Ws, 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.
. 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 resolves the problem:
Tu is called the trace of u. Roughly speaking, this theorem extends the restriction operator to the Sobolev space W1,p(Ω) for well-behaved Ω. Note that the trace operator
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 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 W1,p(Ω) can be approximated by smooth functions with compact support.
is an element of Lp(ℝn). Furthermore,
In the case of the Sobolev space W1,p(Ω), extending a function u by zero will not necessarily yield an element of W1,p(ℝn). But for Ω bounded with Lipschitz boundary, there exists for every 1 ≤ p ≤ ∞ a bounded extension operator
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 space
Vector 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 Lp-norms of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable weak sense
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 C1 — see smooth function
Smooth 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 L2-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 ∈ Ck(Ω), where k is a natural number
Natural 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 ∈ Cc∞(Ω),
,where α a multi-index of order |α| = k and Ω is an open subset in ℝn. 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
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...
of u. If there exists a weak α-th partial derivative of u, then it is uniquely defined almost everywhere
Almost 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 ∈ 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.
The Sobolev spaces Wk,p(Ω) combine the concepts of weak differentiability and Lebesgue norms.
Definition
The Sobolev space Wk,p(Ω) is defined to be the set of all functions u ∈ Lp(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 Lp(Ω), i.e.
Here, Ω is an open set in ℝn and 1 ≤ p ≤ +∞. The natural number
Natural 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 Wk,p(Ω).
There are several choices for a norm for Wk,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, Wk,p(Ω) is a Banach space. For finite p, Wk,p(Ω) is also a separable space. It is conventional to denote Wk,2(Ω) by Hk(Ω) for it is a Hilbert space
Hilbert 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 ∈ Wk,p(Ω) can be approximated by smooth functions. If p is finite and Ω is bounded with Lipschitz boundary, then for any u ∈ Wk,p(Ω) there exists an approximating sequence of functions um ∈ C∞, smooth up to the boundary such that ||um-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 Wk,p(ℝn) 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 (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 Hs,p(ℝn). They are Banach spaces in general and Hilbert spaces in the special case p = 2 .
For an open set Ω ⊆ ℝn, Hs,p(Ω) is the set of restrictions of functions from Hs,p(ℝn) 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 . By 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
the Bessel potential spaces Hs,p(ℝn) form a continuous scale between the Sobolev spaces Wk,p(ℝn). 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
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 Lp-setting. For an open subset Ω of ℝn, , θ ∈ (0,1) and f ∈ Lp(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 Ws, 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 Ws, 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 ∈ Wk,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 W1,p(Ω) for well-behaved Ω. Note that the trace operator
Trace 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 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 W1,p(Ω) can be approximated by smooth functions with compact support.
Extensions
For a function f ∈ Lp(Ω) on an open set Ω ∈ ℝn, its extension by zerois an element of Lp(ℝn). Furthermore,
In the case of the Sobolev space W1,p(Ω), extending a function u by zero will not necessarily yield an element of W1,p(ℝn). But for Ω bounded with Lipschitz boundary, there exists for every 1 ≤ p ≤ ∞ a bounded extension operator
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...
.