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.
Article
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.
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.
Contents
Motivation
Throughout the article,
Ω
{\displaystyle \Omega }
is an open subset of
R
n
.
{\displaystyle \mathbb {R} ^{n}.}
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
{\displaystyle C^{1}}
— 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
C
1
{\displaystyle C^{1}}
(or
C
2
{\displaystyle 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. A typical example is measuring the energy of a temperature or velocity distribution by an
Sobolev spaces with integer k
One-dimensional case
In the one-dimensional case the Sobolev space
W
k
,
p
(
R
)
{\displaystyle W^{k,p}(\mathbb {R} )}
for
1
≤
p
≤
∞
{\displaystyle 1\leq p\leq \infty }
is defined as the subset of functions
f
{\displaystyle f}
in
L
p
(
R
)
{\displaystyle L^{p}(\mathbb {R} )}
such that
f
{\displaystyle f}
and its weak derivatives up to order
k
{\displaystyle k}
Multidimensional case
The transition to multiple dimensions brings more difficulties, starting from the very definition. The requirement that
f
(
k
−
1
)
{\displaystyle f^{(k-1)}}
be the integral of
f
(
k
)
{\displaystyle f^{(k)}}
does not generalize, and the simplest solution is to consider derivatives in the sense of distribution theory.
Sobolev spaces are often considered when investigating partial differential equations. It is essential to consider boundary values of Sobolev functions. If
u
∈
C
(
Ω
)
{\displaystyle u\in C(\Omega )}
, those boundary values are described by the restriction
u
|
∂
Ω
.
{\displaystyle u|_{\partial \Omega }.}
However, it is not clear how to describe values at the boundary for
u
∈
W
k
,
p
(
Ω
)
,
{\displaystyle u\in W^{k,p}(\Omega ),}
as the n-dimensional measure of the boundary is zero. The following theorem resolves the problem:
Sobolev spaces with non-integer k
Bessel potential spaces
For a natural number k and 1 < p < ∞ one can show (by using Fourier multipliers) that the space
W
k
,
p
(
R
n
)
{\displaystyle W^{k,p}(\mathbb {R} ^{n})}
can equivalently be defined as
W
k
,
p
(
R
n
)
=
H
k
,
p
(
R
n
)
:=
{
f
∈
L
p
(
Sobolev–Slobodeckij spaces
Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder condition to the Lp-setting. For
the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by
[
f
]
θ
,
p
Extension operators
If
Ω
{\displaystyle \Omega }
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
Ω
{\displaystyle \Omega }
to functions of
R
n
{\displaystyle \mathbb {R} ^{n}}
such that:
Au(x) = u(x) for almost every x in
Ω
{\displaystyle \Omega }
and
A
:
W
k
,
p
(
Ω
)
→
W
k
,
p
(
R
Case of p = 2
Extension operators are the most natural way to define
H
s
(
Ω
)
{\displaystyle H^{s}(\Omega )}
for non-integer s (we cannot work directly on
Ω
{\displaystyle \Omega }
since taking Fourier transform is a global operation). We define
H
s
(
Ω
)
{\displaystyle H^{s}(\Omega )}
by saying that
u
∈
H
s
(
Ω
)
{\displaystyle u\in H^{s}(\Omega )}
if and only if
A
u
∈
H
s
Extension by zero
Like above, we define
H
0
s
(
Ω
)
{\displaystyle H_{0}^{s}(\Omega )}
to be the closure in
H
s
(
Ω
)
{\displaystyle H^{s}(\Omega )}
of the space
C
c
∞
(
Ω
)
{\displaystyle C_{c}^{\infty }(\Omega )}
of infinitely differentiable compactly supported functions. Given the definition of a trace, above, we may state the following
If
u
∈
H
0
s
(
Ω
)
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
W
k
,
p
{\displaystyle W^{k,p}}
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
W
k
,
∞
{\displaystyle W^{k,\infty }}
is defined to be the Hölder space Cn,α where k = n + α and 0 < α ≤ 1.) The Sobolev embedding theorem states that if
k
⩾
m
{\displaystyle k\geqslant m}
and
k
−
n
p
⩾
m
−
n
Article from Wikipedia (CC BY-SA 4.0), where it is maintained by volunteer editors.
L
2
{\displaystyle 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
)
{\displaystyle u\in C^{k}(\Omega )}
, where
k
{\displaystyle k}
is a natural number, and for all infinitely differentiable functions with compact support
satisfies the definition for being the weak derivative of
u
(
x
)
,
{\displaystyle u(x),}
which then qualifies as being in the Sobolev space
W
1
,
p
{\displaystyle W^{1,p}}
(for any allowed
p
{\displaystyle p}
, see definition below).
The Sobolev spaces
W
k
,
p
(
Ω
)
{\displaystyle W^{k,p}(\Omega )}
combine the concepts of weak differentiability and Lebesgue norms.
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
(
k
−
1
)
{\displaystyle (k{-}1)}
-th derivative
f
(
k
−
1
)
{\displaystyle f^{(k-1)}}
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,
is equivalent to the norm above (i.e., the induced topologies of the norms are the same).
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:
H
k
=
W
k
,
2
.
{\displaystyle H^{k}=W^{k,2}.}
The space
H
k
{\displaystyle H^{k}}
can be defined naturally in terms of Fourier series whose coefficients decay sufficiently rapidly, namely,
In one dimension, some other Sobolev spaces permit a simpler description. For example,
W
1
,
1
(
0
,
1
)
{\displaystyle 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
)
{\displaystyle W^{1,\infty }(I)}
is the space of bounded 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
W
k
,
∞
{\displaystyle W^{k,\infty }}
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
<
∞
.
{\displaystyle p<\infty .}
(E.g., functions behaving like |x|−1/3 at the origin are in
is also a separable space. It is conventional to denote
W
k
,
2
(
Ω
)
{\displaystyle W^{k,2}(\Omega )}
by
H
k
(
Ω
)
{\displaystyle H^{k}(\Omega )}
for it is a Hilbert space with the norm
‖
⋅
‖
W
k
,
2
(
Ω
)
{\displaystyle \|\cdot \|_{W^{k,2}(\Omega )}}
.
It is rather hard to work with Sobolev spaces relying only on their definition. It is therefore interesting to know that by the Meyers–Serrin theorem a function
u
∈
W
k
,
p
(
Ω
)
{\displaystyle u\in W^{k,p}(\Omega )}
can be approximated by smooth functions. This fact often allows us to translate properties of smooth functions to Sobolev functions. If
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
)
{\displaystyle L^{2}(\Omega )}
consisting of eigenvectors of the Laplace operator (with Dirichlet boundary condition).
Tu is called the trace of u. Roughly speaking, this theorem extends the restriction operator to the Sobolev space
W
1
,
p
(
Ω
)
{\displaystyle W^{1,p}(\Omega )}
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
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 a is domain with uniform Ck-boundary, k a natural number and 1 < p < ∞. By the embeddings
form a continuous scale between the Sobolev spaces
W
k
,
p
(
R
n
)
.
{\displaystyle W^{k,p}(\mathbb {R} ^{n}).}
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
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
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
W
1
,
p
(
R
n
)
.
{\displaystyle W^{1,p}(\mathbb {R} ^{n}).}
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
then the embedding is completely continuous (this is sometimes called Kondrachov's theorem or the Rellich–Kondrachov theorem). Functions in
W
m
,
∞
{\displaystyle W^{m,\infty }}
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
R
n
{\displaystyle \mathbb {R} ^{n}}
(Stein 1970). Sobolev embeddings on
R
n
{\displaystyle \mathbb {R} ^{n}}
that are not compact often have a related, but weaker, property of cocompactness.
