User:Silly rabbit/Sobolev space

In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of L^p 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. 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.

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.

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 W^k,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 L^p(D), i.e.

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

There are several choices of norm for W^k,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, W^k,p(D) is a Banach space. For finite p, W^k,p(D) is also a separable space. As noted above, it is conventional to denote W^k,2(D) by H^k(D).

The fractional order Sobolev spaces H^s(Rⁿ), s ≥ 0, can be defined using the Fourier transform as before:

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

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 H^s(D) is given by

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

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.

Examples

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

$f(x)={\frac {1}{|x|^{\alpha }}}$

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.

Sobolev embedding

Main article: Sobolev inequality

Write $W^{k,p}$ 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 $W^{k,\infty }$ is defined to be the Hölder space C^n,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 L^p 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):

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 $C^{m}$, m ≥ s. (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 $H^{s-1/2}(G)$. 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 $W^{s,p}$ 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 $W^{s,p}$ 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 $W^{k,p}(X)$ 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 $H^{s}(X)$ 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 $H_{0}^{s}(X)$ to be the closure in $H^{s}(X)$ of the space $C_{c}^{\infty }(X)$ 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

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

where d/dn is the derivative normal to G, and k is the largest integer less than s. Then $H_{0}^{s}$ is precisely the kernel of P.

If $u\in H_{0}^{s}(X)$ 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.

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) [1994], "Imbedding theorems", Encyclopedia of Mathematics, EMS Press
• Nikol'skii, S.M. (2001) [1994], "Sobolev space", Encyclopedia of Mathematics, EMS Press
• 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

Category:Sobolev spaces Category:Fourier analysis