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In [[mathematics]], a '''Sobolev space''' is a [[vector space]] of functions equipped with a [[normed space|norm]] that is a combination of [[Lp norm|''L<sup>p</sup>'' norms]] of the function itself as well as its derivatives up to a given order. The derivatives are understood in a suitable [[weak derivative|weak sense]] to make the space [[Complete metric 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 [[Russia|Russian]] [[mathematician]] [[Sergei Lvovich Sobolev|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 function]]s and with the [[derivative]]s understood in the classical sense.
== Introduction ==
There are many criteria for smoothness of [[mathematical function]]s. The most basic criterion may be that of [[continuous function|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''<sup>1</sup> — see [[smooth function]]). Differentiable functions are important in many areas, and in particular for [[differential equation]]s. In the twentieth century, however, it was observed that the space ''C''<sup>1</sup> (or ''C''<sup>2</sup>, 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.
== 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 ''W''<sup>k,p</sup> is defined to be the subset of [[Lp space|''L''<sup>p</sup>]] such that function ''f'' and its [[weak derivative]]s up to some order ''k'' have a finite [[Lp norm|''L''<sup>p</sup> 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 the (''k''-1)-th derivative of function ''f'', ''f''<sup>(k-1)</sup>, is differentiable almost everywhere and is equal almost everywhere to the [[Lebesgue integration|Lebesgue integral]] of its derivative (this gets rid of examples such as [[Cantor function|Cantor's function]] which are irrelevant to what the definition is trying to accomplish).
With this definition, the Sobolev spaces admit a natural [[Normed vector space|norm]],
:
Space ''W''<sup>k,p</sup> equipped with the norm <math>\|\cdot\|_{k,p}</math> 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
is [[Normed_vector_space#Distances_in_normed_vector_spaces|equivalent]] to the norm above.
===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:
:<math>\,H^k = W^{k,2}.</math>
The space <math>H^k</math> can be defined naturally in terms of [[Fourier series]] whose coefficients decay sufficiently rapidly, namely,
:<math>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\}</math>
where <math>\widehat{f}</math> is the Fourier series of <math>f</math>. As above, one can use the equivalent norm
:<math>\|f\|^2=\sum_{n=-\infty}^\infty (1 + |n|^{2})^k |\widehat{f}(n)|^2.</math>
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 ''H''<sup>''k''</sup> admits an [[inner product space|inner product]], like the space ''H''<sup>0</sup> = ''L''<sup>2</sup>. In fact, the ''H''<sup>''k''</sup> inner product is defined in terms of the ''L''<sup>2</sup> inner product:
:<math>\langle u,v\rangle_{H^k}=\sum_{i=0}^k\langle D^i u,D^i v\rangle_{L^2}.</math>
The space ''H''<sup>''k''</sup> becomes a Hilbert space with this inner product.
===Other examples===
Some other Sobolev spaces permit a simpler description. For example, <math>W^{1,1}(0,1)</math> is the space of [[absolute continuity|absolutely continuous functions]] on <math>(0,1)</math>, while ''W''<sup>1,∞</sup>(''I'') is the space of [[Lipschitz continuity|Lipschitz functions]] on <math>I</math>, for every interval <math>I</math>.
All spaces ''W''<sup>k,∞</sup> are (normed) [[algebra (ring theory)|algebra]]s, 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''|<sup>−1/3</sup> at the origin are in ''L''<sup>2</sup>, but the product of two such functions is not in ''L''<sup>2</sup>).
===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. <math>W^{s,p}</math> or <math>H^s.</math>
==== The case ''p'' = 2 ====
The case ''p'' = 2 is the easiest since the Fourier description is straightforward to generalize. We define the norm
:<math>\|f\|^2_{s,2}=\sum (1+n^2)^s|\widehat{f}(n)|^2</math>
and the Sobolev space <math>H^s</math> 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
:<math>F^s(f)=\sum_{n=-\infty}^\infty (in)^s\widehat{f}(n)e^{int}</math>
or in other words, taking Fourier transform, multiplying by <math>(in)^s</math> and then taking inverse Fourier transform (operators defined by Fourier-multiplication-inverse Fourier are called [[multiplier (Fourier analysis)|multipliers]] and are a topic of research in their own right). This allows to define the Sobolev norm of <math>s,p</math> by
:<math>\|f\|_{s,p}=\|f\|_p+\|F^s(f)\|_p</math>
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'']<sub>''t''</sub>. (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'']<sub>''a''</sub> , [''X,Y'']<sub>''b''</sub> ]<sub>''c''</sub> = [''X,Y'']<sub>''cb''+(1-''c'')''a''</sub>.
''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'']''<sub>t</sub> to ''[''A,B'']''<sub>t</sub>. and we have the interpolation inequality:''
<math>\|T\|_{[X,Y]_t \to [A,B]_t}\leq C\|T\|_{X\to A}^{1-t}\|T\|_{Y\to B}^t.</math>
See also: [[Riesz-Thorin theorem]].
Returning to Sobolev spaces, we want to get <math>W^{s,p}</math> for non-integer ''s'' by interpolating between <math>W^{k,p}</math>-s. The first thing is of course to see that this gives consistent results, and indeed we have
''Theorem: <math>\left[W^{0,p},W^{m,p}\right]_t=W^{n,p}</math> if n is an integer such that n=tm.''
Hence, complex interpolation is a consistent way to get a continuum of spaces <math>W^{s,p}</math> between the <math>W^{k,p}</math>. Further, it gives the same spaces as fractional order differentiation does (but see [[#Extension operators|extension operators]] below for a twist).
==Multiple dimensions==
We now turn to the case of Sobolev spaces in '''R'''<sup>''n''</sup> and subsets of '''R'''<sup>''n''</sup>. The change from the circle to the [[line (mathematics)|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''<sup>(''k''−1)</sup> is the integral of ''f''<sup>(''k'')</sup> 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 '''R'''<sup>n</sup>, let ''k'' be a [[natural number]] and let 1 ≤ ''p'' ≤ +∞. The Sobolev space ''W''<sup>''k'',''p''</sup>(''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]]
:<math>f^{(\alpha)} = \frac{\partial^{| \alpha |} f}{\partial x_{1}^{\alpha_{1}} \dots \partial x_{n}^{\alpha_{n}}}</math>
is both [[locally integrable function|locally integrable]] and in ''L''<sup>''p''</sup>(''D''), i.e.
:<math>\|f^{(\alpha)}\|_{L^{p}} < \infty.</math>
There are several choices of norm for ''W''<sup>''k'',''p''</sup>(''D''). The following two are common, and are equivalent in the sense of equivalence of norms:
:<math>\| 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}</math>
and
:<math>\| 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}</math>
With respect to either of these norms, ''W''<sup>''k'',''p''</sup>(''D'') is a Banach space. For finite ''p'', ''W''<sup>''k'',''p''</sup>(''D'') is also a [[separable space]]. As noted above, it is conventional to denote ''W''<sup>''k'',2</sup>(''D'') by ''H''<sup>''k''</sup>(''D'').
The fractional order Sobolev spaces ''H''<sup>''s''</sup>('''R'''<sup>''n''</sup>), ''s'' ≥ 0, can be defined using the Fourier transform (using the fact that the Fourier Transform is a unitary transformation) as before:
:<math>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 } \big)^{s} \big| \hat{f} (\xi) \big|^{2} \, \mathrm{d} \xi < + \infty \right. \right\}.</math>
However, if ''D'' is not a periodic domain like '''R'''<sup>''n''</sup> or the torus '''T'''<sup>''n''</sup>, 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 ''L''<sup>2</sup> analogue of [[Hölder-continuous function|Hölder continuity]]: an equivalent inner product for ''H''<sup>''s''</sup>(''D'') is given by
:<math>(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,</math>
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''<sup>1,1</sup> contains only continuous functions. For example, 1/|''x''| belongs to ''W''<sup>1,1</sup>('''B'''<sup>3</sup>) where '''B'''<sup>3</sup> is the unit ball in three dimensions. For ''k'' > ''n''/''p'' the space ''W''<sup>''k'',''p''</sup>(''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'' : '''B'''<sup>''n''</sup> → '''R''' ∪ {+∞} defined on the ''n''-dimensional ball and given by
:<math>f(x) = \frac1{| x |^{\alpha}}</math>
lies in ''W''<sup>''k'',''p''</sup>('''B'''<sup>''n''</sup>) [[if and only if]]
:<math>\alpha < \frac{n}{p} - k.</math>
Intuitively, the blow-up of ''f'' at 0 "counts for less" when ''n'' is large since the unit ball is "smaller" in higher dimensions.
===Absolutely Continuous on Lines (ACL) characterization of Sobolev functions===
Let Ω be an open set in '''R'''<sup>n</sup> and 1 ≤ ''p'' ≤ ∞. If a function is in ''W''<sup>1,''p''</sup>(Ω), then, possibly after modifying the function on a set of measure zero, the restriction to [[almost every]] line parallel to the coordinate directions in '''R'''<sup>''n''</sup> is [[absolutely continuous]]. Conversely, if the restriction of ''ƒ'' to almost every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient ∇''ƒ'' exists [[almost everywhere]], and ''ƒ'' is in ''W''<sup>1,p</sup>(Ω) provided ''ƒ'' and |∇''f''| are both in ''L''<sup>p</sup>(Ω). In particular, in this case the weak partial derivatives of ''ƒ'' and pointwise partial derivatives of ''ƒ'' agree almost everywhere.
A stronger result holds in the case ''p'' = ∞. A function in ''W''<sup>1,∞</sup>(Ω) is, after modifying on a set of measure zero, locally Lipschitz.
=== Functions vanishing at the boundary ===
Let ''Ω''  be an open set in '''R'''<sup>''n''</sup>. The Sobolev space ''W''<sup>1,2</sup>(''Ω'') is also denoted by ''H''<sup>1</sup>(''Ω''). It is a Hilbert space, with an important subspace <math>H^1_0(\Omega)</math>, defined to be the closure in ''H''<sup>1</sup>(''Ω'') of the infinitely differentiable functions compactly supported in ''Ω''. The Sobolev norm defined above reduces here to
:<math>\|f\|_{H^1} = \Bigl( \int_\Omega \bigl( |f|^2 + |\nabla f|^2 \bigr) \Bigr)^{1/2}.</math>
When ''Ω''  has a regular boundary, <math>H^1_0(\Omega)</math> can be described as the space of functions in ''H''<sup>1</sup>(''Ω'') that vanish at the boundary, in the sense of traces ([[Sobolev space#Extension by zero|see below]]). When ''n'' = 1, if ''Ω'' = (''a'', ''b'') is a bounded interval, then <math>H^1_0(a, b)</math> consists of continuous functions on [''a'', ''b''] of the form
:<math>f(x) = \int_a^x f'(t) \, \mathrm{d}t, \quad x \in [a, b]</math>
where the generalized derivative ''ƒ''′ is in ''L''<sup>2</sup>(''a'', ''b'') and has 0 integral, so that ''ƒ''(''b'') = ''ƒ''(''a'') = 0.
When ''Ω''  is bounded, the [[Poincaré inequality]] states that there is a constant ''C'' = ''C''(''Ω'') such that
:<math>\int_\Omega | f|^2 \le C^2 \, \int_\Omega |\nabla f|^2, \quad f \in H^1_0(\Omega).</math>
When ''Ω'' is bounded, the injection from <math>H^1_0(\Omega)</math> to ''L''<sup>2</sup>(''Ω'') is [[Compact operator|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''<sup>2</sup>(''Ω'') consisting of eigenvectors of the [[Laplace operator]] (with [[Dirichlet boundary condition]]).
==Sobolev embedding==
{{main|Sobolev inequality}}
Write <math>W^{k,p}</math> 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 <math>W^{k,\infty}</math> is defined to be the [[Hölder space]] ''C''<sup>''n'',α</sup> where ''k'' = ''n'' + α and 0 < α ≤ 1.) The Sobolev embedding theorem states that
if ''k'' ≥ ''m'' and ''k'' − ''n''/''p'' ≥ ''m'' − ''n''/''q'' then
:<math>W^{k,p}\subseteq W^{m,q}</math>
and the embedding is continuous. Moreover if ''k'' > ''m'' and ''k'' − ''n''/''p'' > ''m'' −''n''/''q''
then the embedding is completely continuous (this is sometimes called '''Kondrakov's theorem'''). Functions in <math>W^{m,\infty}</math> 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 ''L''<sup>''p''</sup> 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'''<sup>''n''</sup> {{harv|Stein|1970}}.
== Traces ==
:''Main article [[Trace operator]].''
Let ''s'' > ½. If ''X'' is an open set such that its [[boundary (topology)|boundary]] ''G'' is "sufficiently smooth", then we may define the ''trace'' (that is, ''restriction'') map ''P'' by
:<math>Pu=u|_G,</math>
i.e. ''u'' restricted to ''G''. A simple smoothness condition is uniformly <math>C^m</math>, ''m'' ≥ ''s''. (There is no connection here to [[trace of a matrix]].)
This trace map ''P'' as defined has domain <math>H^s(X)</math>, and its image is precisely <math>H^{s-1/2}(G)</math>. To be completely formal, ''P'' is first defined for [[infinitely differentiable function]]s and is extended by continuity to <math>H^s(X)</math>. Note that we 'lose half a derivative' in taking this trace.
Identifying the image of the trace map for <math>W^{s,p}</math> is considerably more difficult and demands the tool of [[real interpolation]]. The resulting spaces are the [[Besov space]]s. It turns out that in the case of the <math>W^{s,p}</math> 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 '''R'''<sup>''n''</sup> such that:
# ''Au''(''x'') = ''u''(''x'') for almost every ''x'' in ''X'' and
# ''A'' is continuous from <math>W^{k,p}(X)</math> to <math>W^{k,p}({\mathbb R}^n)</math>, 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 <math>H^s(X)</math> for non-integer ''s'' (we cannot work directly on ''X'' since taking Fourier transform is a global operation). We define <math>H^s(X)</math> by saying that ''u'' is in <math>H^s(X)</math> if and only if ''Au'' is in <math>H^s(\mathbb R^n)</math>. Equivalently, complex interpolation yields the same <math>H^s(X)</math> 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 <math>H^s(X)</math> spaces.
As a result, the interpolation inequality still holds.
=== Extension by zero ===
We define <math>H^s_0(X)</math> to be the closure in <math>H^s(X)</math> of the space <math>C^\infty_c(X)</math> of infinitely differentiable compactly supported functions. Given the definition of a trace, above, we may state the following
''Theorem: Let X be uniformly C<sup>m</sup> regular, m ≥ s and let P be the linear map sending u in <math>H^s(X)</math> to ''
:<math>\left.\left(u,\frac{du}{dn}, \dots, \frac{d^k u}{dn^k}\right)\right|_G</math>
''where d/dn is the derivative normal to G, and k is the largest integer less than s. Then <math>H^s_0</math> is precisely the kernel of P.''
If <math>u\in H^s_0(X)</math> we may define its extension by zero <math>\tilde u \in L^2({\mathbb R}^n)</math> in the natural way, namely
:<math>\tilde u(x)=u(x) \; \textrm{ if } \; x \in X, 0 \; \textrm{ otherwise.}</math>
''Theorem: Let s > ½. The map taking u to <math>\tilde u</math> is continuous into <math>H^s({\mathbb R}^n)</math> 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.
*{{springer|id=i/i050230|title=Imbedding theorems|first=S.M.|last= Nikol'skii}}
*{{springer|id=S/s085980|title=Sobolev space|first=S.M.|last= Nikol'skii}}
*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)
*{{citation|last=Stein|first=E|title= Singular Integrals and Differentiability Properties of Functions, |publisher=Princeton Univ. Press|year=1970| ISBN= 0-691-08079-8}}
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[[Category:Fourier analysis]]
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