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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 space of functions with sufficiently many derivatives for some application domain, such as [[partial differential equation]]s, and equipped with a norm that measures both the size and regularity of a function.
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| Sobolev spaces are named after the Russian [[mathematician]] [[Sergei Lvovich Sobolev|Sergei Sobolev]]. Their importance comes from the fact that solutions of partial differential equations are naturally found in Sobolev spaces, rather than in spaces of [[continuous function]]s and with the [[derivative]]s understood in the classical sense.
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| ==Motivation==
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| 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. On the other hand, quantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms, rather than the [[uniform norm]]. A typical example is measuring the energy of a temperature or velocity distribution by an ''L''<sup>2</sup>-norm. It is therefore important to develop a tool for differentiating [[Lp space#Lp space|Lebesgue space]] functions.
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| The [[integration by parts]] formula yields that for every ''u'' ∈ ''C''<sup>k</sup>(Ω), where ''k'' is a [[natural number]] and for all infinitely differentiable functions with [[compact support]] ''φ'' ∈ ''C''<sub>c</sub><sup>∞</sup>(Ω),
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| :<math> \int_\Omega uD^\alpha\varphi\;dx=(-1)^{|\alpha|}\int_\Omega \varphi D^\alpha u\;dx</math>,
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| where ''α'' a [[multi-index]] of order |''α''| = ''k'' and Ω is an [[open subset]] in ℝ''<sup>n</sup>''. Here, the notation
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| :<math>D^{\alpha}f = \frac{\partial^{| \alpha |} f}{\partial x_{1}^{\alpha_{1}} \dots \partial x_{n}^{\alpha_{n}}},</math>
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| is used.
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| The left-hand side of this equation still makes sense if we only assume ''u'' to be [[locally integrable]]. If there exists a locally integrable function ''v'', such that
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| :<math> \int_\Omega uD^\alpha\varphi\;dx=(-1)^{|\alpha|}\int_\Omega \varphi v \;dx, \ \ \ \ \varphi\in C_c^\infty(\Omega),</math>
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| we call ''v'' the [[weak derivative|weak ''α''-th partial derivative]] of ''u''. If there exists a weak ''α''-th partial derivative of ''u'', then it is uniquely defined [[almost everywhere]].
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| On the other hand, if ''u'' ∈ ''C''<sup>k</sup>(Ω), then the classical and the weak derivative coincide. Thus, if ''v'' is a weak ''α''-th partial derivative of ''u'', we may denote it by ''D''<sup>α</sup>''u'' := ''v''.
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| The Sobolev spaces ''W<sup>k,p</sup>''(Ω) combine the concepts of weak differentiability and [[Lp norm|Lebesgue norms]].
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| ==Sobolev spaces with integer k==
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| ===Definition===
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| The Sobolev space ''W<sup>k,p</sup>''(Ω) is defined to be the set of all functions ''u'' ∈ ''L<sup>p</sup>''(Ω) such that for every [[multi-index]] α with |α| ≤ ''k'', the weak [[partial derivative]] ''D''<sup>α</sup>''u'' belongs to ''L<sup>p</sup>''(Ω), i.e.
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| : <math> W^{k,p}(\Omega) = \left \{ u \in L^p(\Omega) : D^{\alpha}u \in L^p(\Omega) \,\, \forall |\alpha| \leq k \right \}. </math>
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| Here, Ω is an open set in ℝ''<sup>n</sup>'' and 1 ≤ ''p'' ≤ +∞. The [[natural number]] ''k'' is called the order of the Sobolev space ''W<sup>k,p</sup>''(Ω).
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| There are several choices for a norm for ''W<sup>k,p</sup>''(Ω). The following two are common and are equivalent in the sense of [[Norm_(mathematics)#Properties|equivalence of norms]]:
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| :<math>\| u \|_{W^{k, p}(\Omega)} := \begin{cases} \left( \sum_{| \alpha | \leq k} \| D^{\alpha}u \|_{L^{p}(\Omega)}^{p} \right)^{\frac{1}{p}}, & 1 \leq p < + \infty; \\ \max_{| \alpha | \leq k} \| D^{\alpha}u \|_{L^{\infty}(\Omega)}, & p = + \infty; \end{cases}</math>
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| and
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| :<math>\| u \|'_{W^{k, p}(\Omega)} := \begin{cases} \sum_{| \alpha | \leq k} \| D^{\alpha}u \|_{L^{p}(\Omega)}, & 1 \leq p < + \infty; \\ \sum_{| \alpha | \leq k} \| D^{\alpha}u \|_{L^{\infty}(\Omega)}, & p = + \infty. \end{cases}</math>
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| With respect to either of these norms, ''W<sup>k,p</sup>''(Ω) is a Banach space. For finite ''p'', ''W<sup>k,p</sup>''(Ω) is also a [[separable space]]. It is conventional to denote ''W''<sup>''k'',2</sup>(Ω) by ''H<sup>k</sup>''(Ω) for it is a [[Hilbert space]] with the norm <math>\| \cdot \|_{W^{k, 2}(\Omega)} </math>.<ref>{{harvnb|Evans|1998|loc=Chapter 5.2}}</ref>
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| ===Approximation by smooth functions===
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| Many of the properties of the Sobolev spaces cannot be seen directly from the definition. It is therefore interesting to investigate under which conditions a function ''u'' ∈ ''W''<sup>k,p</sup>(Ω) can be approximated by [[smooth functions]]. If ''p'' is finite and Ω is bounded with Lipschitz boundary, then for any ''u'' ∈ ''W''<sup>k,p</sup>(Ω) there exists an approximating sequence of functions ''u''<sub>m</sub> ∈ ''C''<sup>∞</sup>({{overline|Ω}}), smooth up to the boundary such that:<ref name="Adams1975">{{harvnb|Adams|1975}}</ref>
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| :<math> \left \| u_m - u \right \|_{W^{k,p}(\Omega)} \to 0.</math>
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| ==Sobolev spaces with non-integer ''k''==
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| ===Bessel potential spaces===
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| For a natural number ''k'' and {{math|1 < p < ∞}} one can show (by using [[Multiplier (Fourier analysis)|Fourier multipliers]]<ref>{{harvnb|Bergh|Löfström|1976}}</ref><ref name="Triebel1995">{{harvnb|Triebel|1995}}</ref>) that the space ''W<sup>k,p</sup>''(ℝ''<sup>n</sup>'') can equivalently be defined as
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| :<math> W^{k,p}(\mathbb{R}^n) = H^{k,p}(\mathbb{R}^n) := \left \{f \in L^p(\mathbb{R}^n) : \mathcal{F}^{-1}(1+ |\xi|^2)^{\frac{k}{2}}\mathcal{F}f \in L^p(\mathbb{R}^n) \right \} </math>
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| with the norm
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| :<math>\|f\|_{H^{k,p}(\mathbb{R}^n)} := \left \|\mathcal{F}^{-1} \left (1+ |\xi|^2 \right )^{\frac{k}{2}}\mathcal{F}f \right \|_{L^p(\mathbb{R}^n)} </math>.
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| This motivates Sobolev spaces with non-integer order since in the above definition we can replace ''k'' by any real number ''s''. The resulting spaces
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| :<math> H^{s,p}(\mathbb{R}^n) := \left \{f \in L^p(\mathbb{R}^n) : \mathcal{F}^{-1}\left (1+ |\xi|^2 \right )^{\frac{s}{2}}\mathcal{F}f \in L^p(\mathbb{R}^n) \right \} </math>
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| are called Bessel potential spaces<ref>Bessel potential spaces with variable integrability have been independently introduced by Almeida & Samko (A. Almeida and S. Samko, "Characterization of [[Riesz potential|Riesz]] and [[Bessel potential]]s on variable [[Lebesgue space]]s", J. Function Spaces Appl. 4 (2006), no. 2, 113–144) and Gurka, Harjulehto & Nekvinda (P. Gurka, P. Harjulehto and A. Nekvinda: "Bessel potential spaces with variable exponent", Math. Inequal. Appl. 10 (2007), no. 3, 661–676).</ref> (named after [[Friedrich Bessel]]). They are Banach spaces in general and Hilbert spaces in the special case ''p'' = 2.
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| For an open set Ω ⊆ ℝ''<sup>n</sup>'', ''H<sup>s,p</sup>''(Ω) is the set of restrictions of functions from ''H<sup>s,p</sup>''(ℝ''<sup>n</sup>'') to Ω equipped with the norm
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| :<math>\|f\|_{H^{s,p}(\Omega)} := \inf \left \{\|g\|_{H^{s,p}(\mathbb{R}^n)} : g \in H^{s,p}(\mathbb{R}^n), g|_{\Omega} = f \right \} </math>.
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| Again, ''H<sup>s,p</sup>''(Ω) is a Banach space and in the case ''p'' = 2 a Hilbert space.
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| Using extension theorems for Sobolev spaces, it can be shown that also ''W<sup>k,p</sup>''(Ω) = ''H<sup>k,p</sup>''(Ω) holds in the sense of equivalent norms, if Ω is domain with uniform ''C''<sup>k</sup>''-boundary, ''k'' a natural number and {{math|1 < p < ∞}}. By the [[embedding]]s
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| :<math> H^{k+1,p}(\mathbb{R}^n) \hookrightarrow H^{s',p}(\mathbb{R}^n) \hookrightarrow H^{s,p}(\mathbb{R}^n) \hookrightarrow H^{k, p}(\mathbb{R}^n), \quad k \leq s \leq s' \leq k+1 </math>
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| the Bessel potential spaces ''H<sup>s,p</sup>''(ℝ''<sup>n</sup>'') form a continuous scale between the Sobolev spaces ''W<sup>k,p</sup>''(ℝ''<sup>n</sup>''). From an abstract point of view, the Bessel potential spaces occur as complex [[interpolation space]]s of Sobolev spaces, i.e. in the sense of equivalent norms it holds that
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| :<math> \left [ W^{k,p}(\mathbb{R}^n), W^{k+1,p}(\mathbb{R}^n) \right ]_\theta = H^{s,p}(\mathbb{R}^n),</math>
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| where: | |
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| :<math>1 \leq p \leq \infty, \ 0 < \theta < 1, \ s= (1-\theta)k + \theta (k+1)= k+\theta. </math>
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| ===Sobolev–Slobodeckij spaces===
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| Another approach to define fractional order Sobolev spaces arises from the idea to generalize the [[Hölder condition]] to the ''L<sup>p</sup>''-setting.<ref>{{harvnb|Lunardi|1995}}</ref> For an open subset Ω of ℝ''<sup>n</sup>'', {{math|1 ≤ p < ∞}}, θ ∈ (0,1) and ''f'' ∈ ''L<sup>p</sup>''(Ω), the '''Slobodeckij seminorm''' (roughly analogous to the Hölder seminorm) is defined by
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| :<math> [f]_{\theta, p, \Omega} :=\left(\int_{\Omega} \int_{\Omega} \frac{|f(x)-f(y)|^p}{|x-y|^{\theta p + n}} \; dx \; dy\right)^{\frac{1}{p}} </math>.
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| Let {{math|s > 0}} be not an integer and set <math>\theta = s - \lfloor s \rfloor \in (0,1)</math>. Using the same idea as for the [[Holder_space#H.C3.B6lder_spaces|Hölder spaces]], the '''Sobolev–Slobodeckij space'''<ref>In the literature, fractional Sobolev-type spaces are also called ''Aronszajn spaces'', ''Gagliardo spaces'' or ''Slobodeckij spaces'', after the names of the mathematicians who introduced them in the 1950s: [[Nachman Aronszajn|N. Aronszajn]] ("Boundary values of functions with finite [[Dirichlet integral]]", Techn. Report of Univ. of Kansas 14 (1955), 77–94), E. Gagliardo ("Proprietà di alcune classi di funzioni in più variabili", ''Ricerche Mat.'' 7 (1958), 102–137), and L. N. Slobodeckij ("Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations", Leningrad. ''Gos. Ped. Inst. Učep. Zap.'' 197 (1958), 54–112).</ref> ''W<sup>s,p</sup>''(Ω) is defined as
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| :<math> W^{s,p}(\Omega) := \left\{f \in W^{\lfloor s \rfloor, p}(\Omega) : \sup_{|\alpha| = \lfloor s \rfloor} [D^\alpha f]_{\theta, p, \Omega} < \infty \right\} </math>.
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| It is a Banach space for the norm
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| :<math> \|f \| _{W^{s, p}(\Omega)} := \|f\|_{W^{\lfloor s \rfloor,p}(\Omega)} + \sup_{|\alpha| = \lfloor s \rfloor} [D^\alpha f]_{\theta, p, \Omega} </math>.
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| If the open subset Ω is suitably regular in the sense that there exist certain extension operators, then also the Sobolev–Slobodeckij spaces form a scale of Banach spaces, i.e. one has the continuous injections or [[embedding]]s
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| :<math> W^{k+1,p}(\Omega) \hookrightarrow W^{s',p}(\Omega) \hookrightarrow W^{s,p}(\Omega) \hookrightarrow W^{k, p}(\Omega), \quad k \leq s \leq s' \leq k+1 </math>.
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| There are examples of irregular Ω such that ''W''<sup>1,''p''</sup>(Ω) is not even a vector subspace of ''W<sup>s,p</sup>''(Ω) for 0 < ''s'' < 1.
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| From an abstract point of view, the spaces ''W<sup>s,p</sup>''(Ω) coincide with the real [[interpolation space]]s of Sobolev spaces, i.e. in the sense of equivalent norms the following holds:
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| :<math> W^{s,p}(\Omega) = \left (W^{k,p}(\Omega), W^{k+1,p}(\Omega) \right)_{\theta, p} , \quad k \in \mathbb{N}, s \in (k, k+1), \theta = s - \lfloor s \rfloor </math>.
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| Sobolev–Slobodeckij spaces play an important role in the study of traces of Sobolev functions. They are special cases of [[Besov space]]s.<ref name="Triebel1995" />
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| ==Traces==
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| Sobolev spaces are often considered when investigating partial differential equations. It is essential to consider boundary values of Sobolev functions. If ''u'' ∈ ''C''(Ω), those boundary values are described by the restriction <math>u|_{\partial\Omega}</math>. However, it is not clear how to describe values at the boundary for ''u'' ∈ ''W<sup>k,p</sup>''(Ω), as the ''n''-dimensional measure of the boundary is zero. The following theorem<ref name="Adams1975" /> resolves the problem:
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| <blockquote>'''Trace Theorem.''' Assume Ω is bounded with [[Lipschitz boundary]]. Then there exists a bounded linear operator <math>T: W^{1,p}(\Omega)\to L^p(\partial\Omega)</math> such that
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| :<math>\begin{align}
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| Tu &= u|_{\partial\Omega} && u\in W^{1,p}(\Omega)\cap C(\overline{\Omega}) \\
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| \left\|Tu\right\|_{L^p(\partial\Omega)}&\leq c(p,\Omega)\|u\|_{W^{1,p}(\Omega)} && u\in W^{1,p}(\Omega).
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| \end{align}</math>
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| </blockquote>
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| ''Tu'' is called the trace of ''u''. Roughly speaking, this theorem extends the restriction operator to the Sobolev space ''W''<sup>1,''p''</sup>(Ω) for well-behaved Ω. Note that the [[trace operator]] ''T'' is in general not surjective, but for 1 < ''p'' < ∞ it maps onto the Sobolev-Slobodeckij space <math>W^{1-\frac{1}{p},p}(\partial\Omega)</math>.
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| Intuitively, taking the trace costs 1/''p'' of a derivative. The functions ''u'' in ''W''<sup>1,p</sup>(Ω) with zero trace, i.e. ''Tu'' = 0, can be characterized by the equality
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| :<math> W_0^{1,p}(\Omega)= \left \{u\in W^{1,p}(\Omega): Tu=0 \right \},</math>
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| where
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| :<math> W_0^{1,p}(\Omega):= \left \{u\in W^{1,p}(\Omega): \exists \{u_m\}_{m=1}^\infty\subset C_c^\infty(\Omega), \ \textrm{such} \ \textrm{that} \ u_m\to u \ \textrm{in} \ W^{1,p}(\Omega) \right \}.</math>
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| In other words, for Ω bounded with Lipschitz boundary, trace-zero functions in ''W''<sup>1,''p''</sup>(Ω) can be approximated by smooth functions with compact support.
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| ==Extensions==
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| For a function ''f'' ∈ ''L<sup>p</sup>''(Ω) on an open subset Ω of ℝ''<sup>n</sup>'', its extension by zero
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| :<math>Ef := \begin{cases} f & \textrm{on} \ \Omega, \\ 0 & \textrm{otherwise} \end{cases}</math>
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| is an element of ''L<sup>p</sup>''(ℝ''<sup>n</sup>''). Furthermore,
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| :<math>\left\| Ef\right\|_{L^p(\mathbb{R}^n)}=\left\| f\right\|_{L^p(\Omega)}.</math>
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| In the case of the Sobolev space ''W''<sup>1,p</sup>(Ω) for 1 ≤ p ≤ ∞ , extending a function ''u'' by zero will not necessarily yield an element of ''W''<sup>1,p</sup>(ℝ''<sup>n</sup>''). But if Ω is bounded with Lipschitz boundary (e.g. ∂Ω is C''<sup>1</sup>''), then for any bounded open set O such that Ω⊂⊂O (i.e. Ω is compactly contained in O), there exists a bounded linear operator<ref name="Adams1975" />
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| :<math> E: W^{1,p}(\Omega)\rightarrow W^{1,p}(\mathbb{R}^n),</math>
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| such that for each ''u'' ∈ ''W''<sup>1,p</sup>(Ω):
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| ''Eu'' = ''u'' a.e. on Ω,
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| ''Eu'' has compact support within O, and
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| there exists a constant ''C'' depending only on p, Ω, O and the dimension ''n'', such that
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| :<math>\left\| Eu\right\|_{W^{1,p}(\mathbb{R}^n)}\leq C\left\|u\right\|_{W^{1,p}(\Omega)}.</math>
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| We call ''Eu'' an extension of ''u'' to ℝ''<sup>n</sup>''.
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| ==Sobolev embeddings==
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| It is a natural question to ask if a Sobolev function is continuous or even continuously differentiable. Roughly speaking, sufficiently many weak derivatives or large ''p'' result in a classical derivative. This idea is generalized and made precise in the [[Sobolev inequality|Sobolev embedding theorem]].
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| ==Notes==
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| {{reflist|30em}}
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| ==References==
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| * {{Citation | last1=Adams | first1=Robert A. | title=Sobolev Spaces | publisher=[[Academic Press]] | location=Boston, MA | isbn=978-0-12-044150-1 | year=1975}}.
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| * {{Citation | last1=Aubin | first1=Thierry | title=Nonlinear analysis on manifolds. Monge-Ampère equations | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-0-387-90704-8 | id={{MathSciNet | id = 681859}} | year=1982 | volume=252}}.
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| *{{citation|first=Otto|last=Nikodym|authorlink=Otto M. Nikodym|title=Sur une classe de fonctions considérée dans l'étude du problème de Dirichlet|journal=Fund. Math.|volume=21|year=1933|pages=129–150|url=http://minidml.mathdoc.fr/cgi-bin/location?id=00113509}}.
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| *{{citation|first=S.L.|last=Sobolev|title=On a theorem of functional analysis|journal=Transl. Amer. Math. Soc.|issue=2|volume=34|year=1963|pages=39–68}}; translation of Mat. Sb., 4 (1938) pp. 471–497.
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| *{{citation|first=S.L.|last=Sobolev|title=Some applications of functional analysis in mathematical physics|publisher=Amer. Math. Soc.|year=1963}}.
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| *{{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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| *{{Citation | last1=Ziemer | first1=William P. | title=Weakly differentiable functions | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-97017-2 | id={{MathSciNet | id = 1014685}} | year=1989 | volume=120}}.
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| ==External links==
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| * [http://arxiv.org/PS_cache/arxiv/pdf/1104/1104.4345v2.pdf Eleonora Di Nezza, Giampiero Palatucci, Enrico Valdinoci (2011). "Hitchhiker's guide to the fractional Sobolev spaces".]
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| {{DEFAULTSORT:Sobolev Space}}
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| [[Category:Sobolev spaces]]
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| [[Category:Fourier analysis]]
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| [[Category:Fractional calculus]]
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| [[Category:Function spaces]]
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