Sobolev space

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In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its weak derivatives up to some order k the condition of finite L^p norm, for given p ≥ 1. It is named after Sergei L. Sobolev.

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 [C^1] — see smooth function). Differentiable functions are important in many areas, and in particular for differential equations. In the twentieth century, 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.

Technical discussion

We start by introducing Sobolev spaces in the simplest settings, the one-dimensional case on the unit circle. In this case the Sobolev space [W^] is defined to be the subset of L^p such that f and its weak derivatives up to some order k have a finite L^p 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 [f^] 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\|_=\sum_^k \|f^\|_p = \sum_^k \Big(\int |f^(t)|^p\,dt \Big)^.]

[W^] equipped with the norm [\|\cdot\|_] 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^\|_p + \|f\|_p]

is equivalent to the norm above.

Examples

A few Sobolev spaces have simpler descriptions. For example, [W^(0,1)] is the space of absolutely continuous functions on [(0,1)], while W^1,∞(I) is the space of Lipschitz functions on [I], for every interval [I]. 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 L², but the product of two such functions is not in L²).

Further, [W^] can be defined naturally in terms of its Fourier series, namely,

[W^() = \Big\):\sum_^\infty (1+n^2 + \dotsb + n^) |\widehat(n)|^2 < \infty\Big\}]

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

[\|f\|^2=\sum_^\infty (1 + n^2)^k |\widehat(n)|^2.]

Both representations follow easily from Parseval's theorem and the fact that differentiation is equivalent to multiplying the Fourier coefficient by in. This specific case is so important it has a special notation, [H^k]:

[\,H^k = W^.]

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. [W^] or [H^s.]

The case p

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

[||f||^2_=\sum (1+n^2)^s|\widehat(n)|^2]

and the Sobolev space [H^s] as the space of all functions with finite norm.

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_^\infty (in)^s\widehat(n)e^]

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||_=||f||_p+||F^s(f)||_p]

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

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 and 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||_\leq C||T||_^||T||_^t.]

See also: Riesz-Thorin theorem.

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

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

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

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^] is the integral of [f^] 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. We define the Sobolev space

[\,W^(D)]

as the family of functions f defined on D such that for every multi-index [\alpha] with

[|\alpha|\leq k]

we have that [f^] is a function and

[||f^||_p < \infty.]

The appropriate norm to take on it is the sum of those L^p norms over all such α. It is then complete, and so a Banach space.

Actually, this approach works also in one dimension, and is not very different from the one described under fractional order differentiation above.

Examples

In multiple dimensions, it is no longer true that, for example, [W^] contains only continuous functions. For example, 1/|x| belong to [W^(B^3)] where [B^3] is the unit ball in three dimensions. It is true that for k sufficiently large, [W^(D)] will contain only continuous functions, but for which k this is already true depends both on p and on the dimension.

However, the descriptions of W^1,∞ and [W^] above hold, mutatis mutandis.

Sobolev embedding

The Sobolev space [W^(\mathbb^n)] is a subset of [L^p(\mathbb^n)] by definition. A natural question to ask is: are there other L^p spaces which contain [W^(\mathbb^n)]? The following answer admits a simple representation (cf. Stein, E., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press (1970). ISBN 0-691-08079-8):

Theorem: Let [k,n\in\mathbb_] and [1\leq p\leq\infty]. Then the following statements hold:

1. if [\frac>\frac] then [W^(\mathbb^n)\subseteq L^-\frac}}(\mathbb^n)] as sets. Moreover, the inclusion is a bounded operator.
2. if [\frac=\frac] then all functions [f\in W^(\mathbb^n)] with compact support are elements of [L^q(\mathbb^n)] where [q<\infty].

Traces

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 sample smoothness condition is uniformly [C^m], m ≥ s. (NB There is no connection here to trace of a matrix.)

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

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

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

As a result, the interpolation inequality still holds.

Extension by zero

We define [H^s_0(X)] to be the closure in [H^s(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 C^m regular, m ≥ s and let P be the linear map sending u in [H^s(X)] to

[\left.\left(u,\frac,...,\frac\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(^n)] in the natural way, namely

[\tilde u(x)=u(x) \; \textrm \; x \in X, 0 \; \textrm]

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

References

 

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