Sobolev space

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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 Banach space or Hilbert 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 smoothness of a function.

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

Contents

[edit] Introduction

There are many criteria for smoothness of mathematical functions. The most basic criterion may be that of continuity. A considerably 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 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.

[edit] Sobolev spaces on the unit circle

We start by introducing Sobolev spaces in the simplest settings, the one-dimensional case on the unit circle. In this case the Sobolev space Wk,p is defined to be the subset of Lp such that f and its weak derivatives up to some order k have a finite Lp norm, for given p ≥ 1. 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) 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,

\|f\|_{k,p}=\Big(\sum_{i=0}^k \|f^{(i)}\|_p^p\Big)^{1/p} = \Big(\sum_{i=0}^k \int |f^{(i)}(t)|^p\,dt \Big)^{1/p}.

Wk,p equipped with the norm \|\cdot\|_{k,p} is 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

\|f^{(k)}\|_p + \|f\|_p

is equivalent to the norm above.

[edit] The case p = 2

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:

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

The space Hk can be defined naturally in terms of Fourier series whose coefficients decay sufficiently rapidly, namely,

H^k({\mathbb T}) = \Big\{ f\in L^2({\mathbb T}):\sum_{n=-\infty}^\infty (1+n^2 + n^4 + \dotsb + n^{2k}) |\widehat{f}(n)|^2 < \infty\Big\}

where \widehat{f} is the Fourier series of f. As above, one can use the equivalent norm

\|f\|^2=\sum_{n=-\infty}^\infty (1 + |n|^{k})^2 |\widehat{f}(n)|^2.

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 Hk admits an inner product, like the space H0 = L2. In fact, the Hk inner product is defined in terms of the L2 inner product:

\langle u,v\rangle_{H^k}=\sum_{i=0}^k\langle D^i u,D^i v\rangle_{L_2}.

The space Hk becomes a Hilbert space with this inner product.

[edit] Other examples

Some other Sobolev spaces permit a simpler description. For example, W1,1(0,1) is the space of absolutely continuous functions on (0,1), while W1,∞(I) is the space of Lipschitz functions on I, for every interval I. All spaces Wk,∞ 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 L2, but the product of two such functions is not in L2).

[edit] Sobolev spaces with non-integer k

To prevent confusion, when talking about k which is not integer we will usually denote it by s, i.e. Ws,p or Hs.

[edit] The case p = 2

The case p = 2 is the easiest since the Fourier description is straightforward to generalize. We define the norm

\|f\|^2_{s,2}=\sum (1+n^2)^s|\widehat{f}(n)|^2

and the Sobolev space Hs as the space of all functions with finite norm.

[edit] Fractional order differentiation

A similar approach can be used if p is different from 2. In this case Parseval's theorem no longer holds, but differentiation still corresponds to multiplication in the Fourier domain and can be generalized to non-integer orders. Therefore we define an operator of fractional order differentiation of order s by

F^s(f)=\sum_{n=-\infty}^\infty (in)^s\widehat{f}(n)e^{int}

or in other words, taking Fourier transform, multiplying by (in)s and then taking inverse Fourier transform (operators defined by Fourier-multiplication-inverse Fourier are called multipliers and are a topic of research in their own right). This allows to define the Sobolev norm of s,p by

\|f\|_{s,p}=\|f\|_p+\|F^s(f)\|_p

and, as usual, the Sobolev space is the space of functions with finite Sobolev norm.

[edit] Complex interpolation

Another way of obtaining the "fractional Sobolev spaces" is given by complex interpolation. Complex interpolation is a general technique: for any 0 ≤ t ≤ 1 and X and Y Banach spaces that are continuously included in some larger Banach space we may create "intermediate space" denoted [X,Y]t. (below we discuss a different method, the so-called real interpolation method, which is essential in the Sobolev theory for the characterization of traces).

Such spaces X and Y are called interpolation pairs.

We mention a couple of useful theorems about complex interpolation:

Theorem (reinterpolation): [ [X,Y]a , [X,Y]b ]c = [X,Y]cb+(1-c)a.

Theorem (interpolation of operators): if {X,Y} and {A,B} are interpolation pairs, and if T is a linear map defined on X+Y into A+B so that T is continuous from X to A and from Y to B then T is continuous from [X,Y]t to [A,B]t. and we have the interpolation inequality:

\|T\|_{[X,Y]_t \to [A,B]_t}\leq C\|T\|_{X\to A}^{1-t}\|T\|_{Y\to B}^t.

See also: Riesz-Thorin theorem.

Returning to Sobolev spaces, we want to get Ws,p for non-integer s by interpolating between Wk,p-s. The first thing is of course to see that this gives consistent results, and indeed we have

Theorem: \left[W^{0,p},W^{m,p}\right]_t=W^{n,p} if n is an integer such that n=tm.

Hence, complex interpolation is a consistent way to get a continuum of spaces Ws,p between the Wk,p. Further, it gives the same spaces as fractional order differentiation does (but see extension operators below for a twist).

[edit] Multiple dimensions

We now turn to the case of Sobolev spaces in Rn and subsets of Rn. The change from the circle to the line only entails technical changes in the Fourier formulas — basically a change of Fourier series to Fourier transform and sums to integrals. The transition to multiple dimensions brings more difficulties, starting from the very definition. The requirement that f(k−1) is 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 D be an open set in Rn, let k be a natural number and let 1 ≤ p ≤ +∞. The Sobolev space Wk,p(D) is defined to be the set of all functions f defined on D such that for every multi-index α with |α| ≤ k, the mixed partial derivative

f^{(\alpha)} = \frac{\partial^{| \alpha |} f}{\partial x_{1}^{\alpha_{1}} \dots \partial x_{n}^{\alpha_{n}}}

is both locally integrable and in Lp(D), i.e.

\|f^{(\alpha)}\|_{L^{p}} < \infty.

There are several choices of norm for Wk,p(D). The following two are common, and are equivalent in the sense of equivalence of norms:

\| f \|_{W^{k, p}} = \begin{cases} \left( \sum_{| \alpha | \leq k} \| f^{(\alpha)} \|_{L^{p}}^{p} \right)^{1/p}, & 1 \leq p < + \infty; \\ \sum_{| \alpha | \leq k} \| f^{(\alpha)} \|_{L^{\infty}}, & p = + \infty; \end{cases}

and

\| f \|'_{W^{k, p}} = \begin{cases} \sum_{| \alpha | \leq k} \| f^{(\alpha)} \|_{L^{p}}, & 1 \leq p < + \infty; \\ \sum_{| \alpha | \leq k} \| f^{(\alpha)} \|_{L^{\infty}}, & p = + \infty. \end{cases}

With respect to either of these norms, Wk,p(D) is a Banach space. For finite p, Wk,p(D) is also a separable space. As noted above, it is conventional to denote Wk,2(D) by Hk(D).

The fractional order Sobolev spaces Hs(Rn), s ≥ 0, can be defined using the Fourier transform (using the fact that the Fourier Transform is a unitary transformation) as before:

H^{s} (\mathbf{R}^{n}) = \left\{ f \colon \mathbf{R}^{n} \to \mathbf{R} \left| \| f \|_{H^{s}}^{2} = \int_{\mathbf{R}^{n}} \big( 1 + | \xi |^{2 s} \big) \big| \hat{f} (\xi) \big|^{2} \, \mathrm{d} \xi < + \infty \right. \right\}.

However, if D is not a periodic domain like Rn or the torus Tn, this definition is insufficient, since the Fourier transform of a function defined on an aperiodic domain is difficult to define. Fortunately, there is an intrinsic characterization of fractional order Sobolev spaces using what is essentially the L2 analogue of Hölder continuity: an equivalent inner product for Hs(D) is given by

(f, g)_{H^{s} (D)} = (f, g)_{H^{k} (D)} + \sum_{| \alpha | = k} \int_{D} \int_{D} \frac{\big( f^{(\alpha)} (x) - f^{(\alpha)} (y) \big) \big( g^{(\alpha)} (x) - g^{(\alpha)} (y) \big)}{| x - y |^{n + 2 t}} \, \mathrm{d} x \mathrm{d} y,

where s = k + t, k an integer and 0 < t < 1. Note that the dimension of the domain, n, appears in the above formula for the inner product.

[edit] Examples

In higher dimensions, it is no longer true that, for example, W1,1 contains only continuous functions. For example, 1/|x| belongs to W1,1(B3) where B3 is the unit ball in three dimensions. For k > n/p the space Wk,p(D) 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, the function f : Bn → R ∪ {+∞} defined on the n-dimensional ball and given by

f(x) = \frac1{| x |^{\alpha}}

lies in Wk,p(Bn) if and only if

\alpha < \frac{n}{p} - k.

Intuitively, the blow-up of f at 0 "counts for less" when n is large since the unit ball is "smaller" in higher dimensions.

[edit] ACL characterization of Sobolev functions

Let Ω be an open set in Rn and 1≤p≤∞. If a function is in W1,p(Ω), then the restriction to every line parallel to the coordinate directions in Rn is absolutely continuous. Conversely, if the restriction of f to every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient ∇f exists almost everywhere, and f is in W1,p(Ω) provided f and |∇f| are both in Lp(Ω). In particular, in this case the weak partial derivatives of f and pointwise partial derivatives of f agree almost everywhere.

[edit] Sobolev embedding

Main article: Sobolev inequality

Write Wk,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,\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≥ l and k−n/p ≥ l−n/q then

W^{k,p}\subseteq W^{l,q}

and the embedding is continuous. Moreover if k> l and k−n/p > l−n/q then the embedding is completely continuous (this is sometimes called Kondrakov's theorem). Functions in W^{l,\infty} have all derivatives of order less than l 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 Rn (Stein 1970):

[edit] Traces

Main article Trace operator.

Let s > ½. If X is an open set such that its boundary G is "sufficiently smooth", then we may define the trace (that is, restriction) map P by

Pu = u | G,

i.e. u restricted to G. A simple smoothness condition is uniformly Cm, m ≥ s. (There is no connection here to trace of a matrix.)

This trace map P as defined has domain Hs(X), and its image is precisely Hs − 1 / 2(G). To be completely formal, P is first defined for infinitely differentiable functions and is extended by continuity to Hs(X). Note that we 'lose half a derivative' in taking this trace.

Identifying the image of the trace map for Ws,p is considerably more difficult and demands the tool of real interpolation. The resulting spaces are the Besov spaces. It turns out that in the case of the Ws,p spaces, we don't lose half a derivative; rather, we lose 1/p of a derivative.

[edit] 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 but more obscure "cone condition") then there is an operator A mapping functions of X to functions of Rn such that:

  1. Au(x) = u(x) for almost every x in X and
  2. A is continuous from Wk,p(X) to W^{k,p}({\mathbb R}^n), for any 1 ≤ p ≤ ∞ and integer k.

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

Extension operators are the most natural way to define Hs(X) for non-integer s (we cannot work directly on X since taking Fourier transform is a global operation). We define Hs(X) by saying that u is in Hs(X) if and only if Au is in H^s(\mathbb R^n). Equivalently, complex interpolation yields the same Hs(X) 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 Hs(X) spaces.

As a result, the interpolation inequality still holds.

[edit] Extension by zero

We define H^s_0(X) to be the closure in Hs(X) of the space C^\infty_c(X) of infinitely differentiable compactly supported functions. Given the definition of a trace, above, we may state the following

Theorem: Let X be uniformly Cm regular, m ≥ s and let P be the linear map sending u in Hs(X) to

\left.\left(u,\frac{du}{dn},...,\frac{d^k u}{dn^k}\right)\right|_G

where d/dn is the derivative normal to G, and k is the largest integer less than s. Then H^s_0 is precisely the kernel of P.

If u\in H^s_0(X) we may define its extension by zero \tilde u \in L^2({\mathbb R}^n) in the natural way, namely

\tilde u(x)=u(x) \; \textrm{ if } \; x \in X, 0 \; \textrm{ otherwise.}

Theorem: Let s>½. The map taking u to \tilde u is continuous into H^s({\mathbb R}^n) if and only if s is not of the form n+½ for n an integer.

[edit] References

  • R.A. Adams, J.J.F. Fournier, 2003. Sobolev Spaces. Academic Press.
  • L.C. Evans, 1998. Partial Differential Equations. American Mathematical Society.
  • Nikol'skii, S.M. (2001), “Imbedding theorems”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
  • Nikol'skii, S.M. (2001), “Sobolev space”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 
  • S.L. Sobolev, "On a theorem of functional analysis" Transl. Amer. Math. Soc. (2) , 34 (1963) pp. 39–68 Mat. Sb. , 4 (1938) pp. 471–497
  • S.L. Sobolev, "Some applications of functional analysis in mathematical physics" , Amer. Math. Soc. (1963)
  • Stein, E (1970), Singular Integrals and Differentiability Properties of Functions,, Princeton Univ. Press, ISBN 0-691-08079-8