User:Temurjin/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.

Sobolev spaces are named after the Russian mathematician Sergei L. Sobolev, who introduced them in 1930s along with a theory of generalized functions. 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 Euclidean space

We start by introducing Sobolev spaces in a simple setting, on the n-dimensional Euclidean space. In this case the Sobolev space W k,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 case it is enough to assume 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). However, for more than one dimensions, this does not generalize, and the simplest solution is to consider derivatives in the sense of distribution theory.

A formal definition now follows. Let k be a natural number and let 1 ≤ p ≤ +∞. The Sobolev space W k,p(Rn) is defined to be the set of all functions f defined on Rn 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(Rn), i.e.

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

We equip the space W k,p(Rn) with the following norm:

\| 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}

With respect to this norm, W k,p(Rn) is a Banach space. For finite p, W k,p(Rn) is also a separable space, and for 1 < p < ∞, these spaces are reflexive.

We have the natural convention W 0,p  =  Lp.

[edit] Equivalent norms

The following is an equivalent norm on the Sobolev space W k,p(Rn):

\| 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}

It turns out that it is enough to take only the first and the highest order terms in the sequence, i.e., the norm defined by

\|f\|_{L^p}+\sum_{|\alpha|=k}\| f^{(\alpha)} \|_{L^{p}},

is equivalent to the norm above.

[edit] Hilbertian Sobolev spaces

Sobolev spaces with p = 2 are especially important because they form a Hilbert space and because of their connection with Fourier series.

The space W k,2 admits an inner product, like the space W 0,2  =  L2. In fact, the W k,2 inner product is defined in terms of the L2 inner product:

\langle u,v\rangle_{W^{k,2}}=\sum_{i=0}^k\langle D^i u,D^i v\rangle_{L_2}.

The space W k,2 becomes a Hilbert space with this inner product.

Furthermore, the space W k,2 can be characterized in terms of Fourier series as follows. For any nonnegative integer k, let us define the space

H^{k} (\mathbf{R}^{n}) = \left\{ f \in\mathcal{S}' : \| f \|_{H^{k}}^{2} = \int_{\mathbf{R}^{n}} \big( 1 + | \xi |^{2 k} \big) \big| \hat{f} (\xi) \big|^{2} \, \mathrm{d} \xi < + \infty \right\},

where \mathcal{S}' is the space of tempered distributions, and \hat{f} is the Fourier series of f. Then from Parseval's theorem and the fact that differentiation is equivalent to multiplying the Fourier coefficient by ξ, it easily follows that

Hk = Wk,2,

with equivalent norms. Evidently, the inner product leading to the norm \|\cdot\|_{H^k} is:

( f,g )_{H^{k}} = \int_{\mathbf{R}^{n}} \big( 1 + | \xi |^{2 k} \big) \hat{f} (\xi) \hat{g} (\xi) \, \mathrm{d} \xi,

with respect to which H k (and therefore W k,2) is a Hilbert space.

[edit] Fractional order Sobolev spaces: p=2

The above Fourier analytic description is straightforward to generalize not only to the case when k is not a natural number, but also to the case k<0. This makes sense in any case. To prevent confusion, when talking about k which may be not integer we will usually use s instead, i.e. H s. For any real s, the fractional Sobolev spaces H s(Rn) can be defined using the Fourier transform as follows:

H^{s} (\mathbf{R}^{n}) = \left\{ f \in\mathcal{S}': \| 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\}.

There is an intrinsic characterization of fractional order Sobolev spaces using what is essentially the L2 analogue of Hölder continuity: an equivalent norm for H s(Rn) is given by

\left(\|f\|_{H^{s}}'\right)^2 = \|f\|_{B^{s,2}}^2 = \|f\|_{W^{k,2}}^2 + \sum_{| \alpha | = k} \int_{\mathbf{R}^n} \int_{\mathbf{R}^n} \frac{| f^{(\alpha)} (x) - f^{(\alpha)} (y) |^2}{| x - y |^{n + 2 t}} \, \mathrm{d} x \mathrm{d} y,

where s = k + t, k an integer and 0 < t < 1. This norm is a special case of the so called Aronszajn-Slobodeckij norm (see below). The reason why we sticked the notation B s,2 in the norm will clear in the next subsection. Note that the dimension of the domain, n, appears in the above formula for the norm.

The spaces H s are all Hilbert spaces, and explicit formulas for inner products can be easily seen from the above formulas for the norms.

[edit] Fractional order Sobolev spaces: General p

The above defined H s-norm can be written in the form

\| f \|_{H^{s}} = \left\| ( 1 + | \cdot |^{2 s} )^{s/2} \hat{f} \right\|_{L^2}= \left\| (( 1 + | \cdot |^{2 s} )^{s/2} \hat{f} )^{\vee}\right\|_{L^2},

where we denoted by (\cdot)^{\vee} the inverse Fourier transform. This gives a hint to define the following more general spaces

H^{s,p} (\mathbf{R}^{n}) = \left\{ f \in\mathcal{S}' : \| f \|_{H^{s,p}} = \left\| (( 1 + | \cdot |^{2 s} )^{s/2} \hat{f} )^{\vee}\right\|_{L^p} < + \infty \right\},

for 1 < p < ∞ and real s. The above definition goes back to N.Aronszajn, K.T.Smith, and A.P.Calderon. The spaces H s,p go by many different names, including fractional Sobolev spaces, Bessel potential spaces, Liouville spaces, and Lebesgue spaces. These are a precursor to the Triebel-Lizorkin spaces. Most importantly, we have

H^{k,p}(\mathbf{R}^n)=W^{k,p}(\mathbf{R}^n),

for all nonnegative integers k and for all 1 < p < ∞, which makes them a very strong candidate for the natural definition of Sobolev spaces for non-integer k. The recent custom seems to be to refer to them as (fractional) Sobolev spaces.

On the other hand, there is another possibility; namely, the above mentioned Aronszajn-Slobodeckij norm has an adaptation to general p:

\|f\|_{B^{s,p}}^p = \|f\|_{W^{k,p}}^p + \sum_{| \alpha | = k} \int_{\mathbf{R}^n} \int_{\mathbf{R}^n} \frac{\left| f^{(\alpha)} (x) - f^{(\alpha)} (y) \right|^p}{| x - y |^{n + p t}} \, \mathrm{d} x \mathrm{d} y,

where 1 < p < ∞, s = k + t > 0, k an integer and 0 < t < 1. Note that in this definition s is always non-integer. We define the space B s,p(Rn) to be the space of tempered distributions for which the above defined B s,p-norm is finite. We know that B s,p = H s,p for p = 2. But this identification stops just there; as long as p is different than 2, B s,p does not coincide with H s,p. The spaces B s,p are called (Besov-) Sobolev spaces, Slobodeckij spaces, or special Besov spaces. They are a precursor to the more general Besov spaces and introduced by N.Aronszajn, L.N.Slobodeckij, and E.Gagliardo as spaces filling gaps between Lp, W 1,p, W 2,p, etc.. Some authors call B s,p Sobolev spaces and denote them simply by W s,p for noninteger s. Together with the original definition of W k,p for integer k, this provides a whole scale of spaces parameterized by real s > 0.

In conclusion, there are two major ways to extend the definition of Sobolev spaces to non-integer k, which differ for p not equal to 2; and one has to be aware of which spaces are being considered in the particular context. As to which one should be considered the "legitimate" way, whereas the recent custom seems to be settled on calling H s,p the Sobolev spaces, in earlier texts one often finds the spaces B s,p are being called Sobolev spaces for non-integer s. Note again that the two definitions are equivalent if p=2.

[edit] Negative order Sobolev spaces: Duality

An important fact about the spaces H s,p is that the topological dual of H s,p is H -s,q for 1 < p < ∞ and for any real s, with q defined by 1/q + 1/p = 1. One can turn this around and use this property as the definition of the spaces W k,p for negative integers k: For integer k > 0 and 1 < p < ∞, with q as above, we define W -k,p(Rn) = [W k,q(Rn)]*. The space W -k,p(Rn) is a Banach space with the norm

\bigl\|u\bigr\|_{W^{-k,p}}=\sup_{v\in W^{k,q}(\mathbf{R}^n)\setminus\{0\}}\frac{|\langle u,v\rangle|}{\|v\|_{W^{k,q}}}

One can characterize the elements of the Sobolev space W -k,p(Rn) as precisely those distributions u\in \mathcal{S}' that can be written as

u=\sum_{|\alpha|\leq k}\partial^\alpha u_{\alpha},

for some functions u_\alpha\in L_p(\mathbf{R}^n). Here all the derivatives are calculated in the sense of distributions.

For any integer k > 0, u\in W^{-k,p}(\mathbf{R}^n) defines a linear operator on v\in W^{k,q}(\mathbf{R}^n) and vice versa by

\langle u,v\rangle=\sum_{|\alpha|\leq k}\langle \partial^\alpha u_{\alpha},v\rangle=\sum_{|\alpha|\leq k}(-1)^{|\alpha|} \langle u_{\alpha},\partial^{\alpha}v\rangle=\sum_{|\alpha|\leq k}(-1)^{|\alpha|} \int_{\mathbf{R}^n} u_{\alpha}\overline{\partial^{\alpha}v} dx

For any integer k, the partial derivative \partial^\alpha is a bounded linear operator from Wk,p to Wk − | α | ,p

[edit] Examples

Some Sobolev spaces permit a simpler description. For example, W 1,1(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).

In higher dimensions, it is no longer true that, for example, W 1,1 contains only continuous functions. For example, 1/|x| belongs to W 1,1(B3) where B3 is the unit ball in three dimensions. For k > n/p the space W k,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 W k,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] 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] Sobolev spaces on domains

Sobolev space of functions acting from \Omega\subseteq \mathbb{R}^n into \mathbb{C} is a generalization of the space of smooth functions, Ck(Ω), by using a broader notion of weak derivatives. In some sense, Sobolev space is a completion of Ck(Ω) under a suitable norm, see Meyers-Serrin Theorem below.

Sobolev spaces are subspaces of the space of integrable functions Lp(Ω) with a certain restriction on their smoothness, that is, their weak derivatives up to a certain order are also integrable functions.

W^{k,p}(\Omega)=\{u\in L_p(\Omega):\partial^{\alpha} u\in L_p(\Omega) for all multi-indeces α such that |\alpha|\leq k\}

This is an original definition, used by Sergei Sobolev.

This space is a Banach space with a norm

\bigl\|u\bigr\|_{k,p,\Omega}^p=\sum_{|\alpha|\leq k} \bigl\|\partial^\alpha u\bigr\|_{L_p}^p =\int_\Omega \sum_{|\alpha|\leq k} |\partial^\alpha u|^p dx

[edit] Density results

For open set \Omega\subset R^n, and for p\in[1,\infty), Ck(Ω) is dense in Wk,p(Ω), that is the Sobolev spaces can alternatively be defined as closure of Ck(Ω), because

W^{k,p}(\Omega)=\operatorname{cl}_{L_p(\Omega),\|\cdot\|_{k,p,\Omega}}\left(\bigl\{f\in C^k(\Omega):\|f\|_{k,p,\Omega}<\infty\bigr\}\right)

Besides, C^k(\overline \Omega) is dense in Wk,p(Ω), if Ω satisfies the so called segment property (in particular if it has Lipschitz boundary).

Note that Ck(Ω) is not dense in W^{k,\infty}(\Omega) because

\operatorname{cl}_{L_\infty(\Omega),\|\cdot\|_{k,\infty,\Omega}}\left(\bigl\{f\in C^k(\Omega):\|f\|_{k,\infty,\Omega}<\infty\bigr\}\right)=C^k(\Omega)

[edit] Negative order spaces

For natural k, the Sobolev spaces W − k,p(Ω) are defined as dual spaces \left(W^{k,q}_0(\Omega)\right)^*, where q is conjugate to p, \frac 1p+\frac 1q =1. Their elements are no longer regular functions, but rather distributions. Alternative definition of Sobolev spaces with negative index is

W^{-k,p}(\Omega)=\left\{u\in D'(\Omega):u=\sum_{|\alpha|\leq k}\partial^\alpha u_{\alpha}, {\rm\ for\ some\ }u_\alpha\in L_p(\Omega)\right\}

Here all the derivatives are calculated in a sense of distributions in space D'(Ω).

These definitions are equivalent. For a natural k, u\in W^{-k,p}(\Omega) defines a linear operator on v\in W^{k,q}_0(\Omega) and vice versa by

\bigl\langle u,v\bigr\rangle=\sum_{|\alpha|\leq k}\bigl\langle \partial^\alpha u_{\alpha},v\bigr\rangle=\sum_{|\alpha|\leq k}(-1)^{|\alpha|} \bigl\langle u_{\alpha},\partial^{\alpha}v\bigr\rangle=\sum_{|\alpha|\leq k}(-1)^{|\alpha|} \int_\Omega u_{\alpha}\overline{\partial^{\alpha}v} dx

Naturally, W − k,p(Ω) is a Banach space with a norm

\bigl\|u\bigr\|_{-k,p,\Omega}=\sup_{v\in W^{k,q}(\Omega),\|v\|_{k,q,\Omega}\not =0}\frac{|\langle u,v\rangle|}{\|v\|_{k,q,\Omega}}

Now for any integer k, \partial^\alpha is a bounded operator from Wk,p to Wk − | α | ,p

[edit] Extensions

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] Sobolev spaces on manifolds

[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] Interpolation of Sobolev spaces

[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] Real interpolation: Besov spaces

[edit] References