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Inmathematics, aSobolev space is avector space of functions equipped with anorm that is a combination ofL^p-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitableweak sense to make the spacecomplete, i.e. aBanach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such aspartial differential equations, and equipped with a norm that measures both the size and regularity of a function.

Sobolev spaces are named after the RussianmathematicianSergei Sobolev. Their importance comes from the fact thatweak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces ofcontinuous functions with thederivatives understood in the classical sense.

In this section and throughout the article${\displaystyle \mathrm{\Omega }}$ is anopen subset of${\displaystyle {\mathbb{R}}^n.}$

There are many criteria for smoothness ofmathematical functions. The most basic criterion may be that ofcontinuity. A stronger notion of smoothness is that ofdifferentiability (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${\displaystyle C^1}$ — seeDifferentiability classes). Differentiable functions are important in many areas, and in particular fordifferential equations. In the twentieth century, however, it was observed that the space${\displaystyle C^1}$ (or${\displaystyle C^2}$, 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.

Quantities or properties of the underlying model of the differential equation are usually expressed in terms of integral norms. A typical example is measuring the energy of a temperature or velocity distribution by an${\displaystyle L^2}$-norm. It is therefore important to develop a tool for differentiatingLebesgue space functions.

Theintegration by parts formula yields that for every${\displaystyle u\in C^k(\mathrm{\Omega })}$, where${\displaystyle k}$ is anatural number, and for all infinitely differentiable functions withcompact support${\displaystyle \phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega }),}$

${\displaystyle {\int }_{\mathrm{\Omega }}u{D}^{\alpha }\phi dx=(-1{)}^{|\alpha |}{\int }_{\mathrm{\Omega }}\phi D^{\alpha }udx,}$

where${\displaystyle \alpha =({\alpha }_1,...,{\alpha }_n)}$ is amulti-index of order${\displaystyle |\alpha |=k}$ and we are using the notation:

${\displaystyle D^{\alpha }f=\frac{{\mathrm{\partial }}^{|\alpha |}f}{\mathrm{\partial }{x}_1^{{\alpha }_1}\dots \mathrm{\partial }{x}_n^{{\alpha }_n}}.}$

The left-hand side of this equation still makes sense if we assume${\displaystyle u}$ to be onlylocally integrable. If there exists a locally integrable function${\displaystyle v}$, such that

${\displaystyle {\int }_{\mathrm{\Omega }}u{D}^{\alpha }\phi dx=(-1{)}^{|\alpha |}{\int }_{\mathrm{\Omega }}\phi vdx\text{for all }\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega }),}$

then we call${\displaystyle v}$ theweak${\displaystyle \alpha }$-th partial derivative of${\displaystyle u}$. If there exists a weak${\displaystyle \alpha }$-th partial derivative of${\displaystyle u}$, then it is uniquely definedalmost everywhere, and thus it is uniquely determined as an element of aLebesgue space. On the other hand, if${\displaystyle u\in C^k(\mathrm{\Omega })}$, then the classical and the weak derivative coincide. Thus, if${\displaystyle v}$ is a weak${\displaystyle \alpha }$-th partial derivative of${\displaystyle u}$, we may denote it by${\displaystyle D^{\alpha }u:=v}$.

For example, the function

is not continuous at zero, and not differentiable at −1, 0, or 1. Yet the function

satisfies the definition for being the weak derivative of${\displaystyle u(x),}$ which then qualifies as being in the Sobolev space${\displaystyle W^{1,p}}$ (for any allowed${\displaystyle p}$, see definition below).

The Sobolev spaces${\displaystyle W^{k,p}(\mathrm{\Omega })}$ combine the concepts of weak differentiability andLebesgue norms.

In the one-dimensional case the Sobolev space${\displaystyle W^{k,p}(\mathbb{R})}$ for${\displaystyle 1\le p\le \mathrm{\infty }}$ is defined as the subset of functions${\displaystyle f}$ in${\displaystyle L^p(\mathbb{R})}$ such that${\displaystyle f}$ and itsweak derivatives up to order${\displaystyle k}$ have a finiteL^p norm. As mentioned above, some care must be taken to define derivatives in the proper sense. In the one-dimensional problem it is enough to assume that the${\displaystyle (k-1)}$-th derivative${\displaystyle f^{(k-1)}}$ is differentiable almost everywhere and is equal almost everywhere to theLebesgue integral of its derivative (this excludes irrelevant examples such asCantor's function).

With this definition, the Sobolev spaces admit a naturalnorm,

${\displaystyle \Vert f{\Vert }_{k,p}={\left(\sum _{i=0}^k{\Vert f^{(i)}\Vert }_p^p\right)}^{\frac{1}{p}}={\left(\sum _{i=0}^k\int {\left|f^{(i)}(t)\right|}^{p}dt\right)}^{\frac{1}{p}}.}$

One can extend this to the case${\displaystyle p=\mathrm{\infty }}$, with the norm then defined using theessential supremum by

${\displaystyle \Vert f{\Vert }_{k,\mathrm{\infty }}=\underset{i=0,\dots ,k}{max}{\Vert f^{(i)}\Vert }_{\mathrm{\infty }}=\underset{i=0,\dots ,k}{max}\left(\text{ess}\underset{t}{sup}\left|f^{(i)}(t)\right|\right).}$

Equipped with the norm${\displaystyle \Vert \cdot {\Vert }_{k,p},W^{k,p}}$ becomes aBanach space. It turns out that it is enough to take only the first and last in the sequence, i.e., the norm defined by

${\displaystyle {\Vert f^{(k)}\Vert }_p+\Vert f{\Vert }_p}$

is equivalent to the norm above (i.e., theinduced topologies of the norms are the same).

Sobolev spaces withp = 2 are especially important because of their connection withFourier series and because they form aHilbert space. A special notation has arisen to cover this case, since the space is a Hilbert space:

${\displaystyle H^k=W^{k,2}.}$

The space${\displaystyle H^k}$ can be defined naturally in terms ofFourier series whose coefficients decay sufficiently rapidly, namely,

where${\displaystyle \hat{f}}$ is the Fourier series of${\displaystyle f,}$ and${\displaystyle \mathbb{T}}$ denotes the 1-torus. As above, one can use the equivalent norm

${\displaystyle \Vert f{\Vert }_{k,2}^2=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{\left(1+|n{|}^2\right)}^k{\left|\hat{f}(n)\right|}^2.}$

Both representations follow easily fromParseval's theorem and the fact that differentiation is equivalent to multiplying the Fourier coefficient by${\displaystyle in}$.

Furthermore, the space${\displaystyle H^k}$ admits aninner product, like the space${\displaystyle H^0=L^2.}$ In fact, the${\displaystyle H^k}$ inner product is defined in terms of the${\displaystyle L^2}$ inner product:

${\displaystyle ⟨u,v{⟩}_{H^k}=\sum _{i=0}^k{⟨D^{i}u,D^{i}v⟩}_{L^2}.}$

The space${\displaystyle H^k}$ becomes a Hilbert space with this inner product.

In one dimension, some other Sobolev spaces permit a simpler description. For example,${\displaystyle W^{1,1}(0,1)}$ is the space ofabsolutely continuous functions on(0, 1) (or rather, equivalence classes of functions that are equal almost everywhere to such), while${\displaystyle W^{1,\mathrm{\infty }}(I)}$ is the space of boundedLipschitz functions onI, for every intervalI. However, these properties are lost or not as simple for functions of more than one variable.

All spaces${\displaystyle W^{k,\mathrm{\infty }}}$ 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 (E.g., functions behaving like |x|^−1/3 at the origin are in${\displaystyle L^2,}$ but the product of two such functions is not in${\displaystyle L^2}$).

The transition to multiple dimensions brings more difficulties, starting from the very definition. The requirement that${\displaystyle f^{(k-1)}}$ be the integral of${\displaystyle f^{(k)}}$ does not generalize, and the simplest solution is to consider derivatives in the sense ofdistribution theory.

A formal definition now follows. Let${\displaystyle k\in \mathbb{N},1⩽p⩽\mathrm{\infty }.}$ The Sobolev space${\displaystyle W^{k,p}(\mathrm{\Omega })}$ is defined to be the set of all functions${\displaystyle f}$ on${\displaystyle \mathrm{\Omega }}$ such that for everymulti-index${\displaystyle \alpha }$ with${\displaystyle |\alpha |⩽k,}$ the mixedpartial derivative

${\displaystyle f^{(\alpha )}=\frac{{\mathrm{\partial }}^{|\alpha |}f}{\mathrm{\partial }{x}_1^{{\alpha }_1}\dots \mathrm{\partial }{x}_n^{{\alpha }_n}}}$

exists in theweak sense and is in${\displaystyle L^p(\mathrm{\Omega }),}$ i.e.

That is, the Sobolev space${\displaystyle W^{k,p}(\mathrm{\Omega })}$ is defined as

${\displaystyle W^{k,p}(\mathrm{\Omega })=\left\{u\in L^p(\mathrm{\Omega }):D^{\alpha }u\in L^p(\mathrm{\Omega })\mathrm{\forall }|\alpha |⩽k\right\}.}$

Thenatural number${\displaystyle k}$ is called the order of the Sobolev space${\displaystyle W^{k,p}(\mathrm{\Omega }).}$

There are several choices for a norm for${\displaystyle W^{k,p}(\mathrm{\Omega }).}$ The following two are common and are equivalent in the sense ofequivalence of norms:

and

With respect to either of these norms,${\displaystyle W^{k,p}(\mathrm{\Omega })}$ is a Banach space. For is also aseparable space. It is conventional to denote${\displaystyle W^{k,2}(\mathrm{\Omega })}$ by${\displaystyle H^k(\mathrm{\Omega })}$ for it is aHilbert space with the norm${\displaystyle \Vert \cdot {\Vert }_{W^{k,2}(\mathrm{\Omega })}}$.^[1]

It is rather hard to work with Sobolev spaces relying only on their definition. It is therefore interesting to know that by theMeyers–Serrin theorem a function${\displaystyle u\in W^{k,p}(\mathrm{\Omega })}$ can be approximated bysmooth functions. This fact often allows us to translate properties of smooth functions to Sobolev functions. If${\displaystyle p}$ is finite and${\displaystyle \mathrm{\Omega }}$ is open, then there exists for any${\displaystyle u\in W^{k,p}(\mathrm{\Omega })}$ an approximating sequence of functions${\displaystyle u_m\in C^{\mathrm{\infty }}(\mathrm{\Omega })}$ such that:

${\displaystyle {\Vert u_m-u\Vert }_{W^{k,p}(\mathrm{\Omega })}\to 0.}$

If${\displaystyle \mathrm{\Omega }}$ hasLipschitz boundary, we may even assume that the${\displaystyle u_m}$ are the restriction of smooth functions with compact support on all of${\displaystyle {\mathbb{R}}^n.}$^[2]

In higher dimensions, it is no longer true that, for example,${\displaystyle W^{1,1}}$ contains only continuous functions. For example,${\displaystyle |x{|}^{-1}\in W^{1,1}({\mathbb{B}}^3)}$ where${\displaystyle {\mathbb{B}}^3}$ is theunit ball in three dimensions. For${\displaystyle k>n/p}$, the space${\displaystyle W^{k,p}(\mathrm{\Omega })}$ will contain only continuous functions, but for which${\displaystyle k}$ this is already true depends both on${\displaystyle p}$ and on the dimension. For example, as can be easily checked usingspherical polar coordinates for the function${\displaystyle f:{\mathbb{B}}^n\to \mathbb{R}\cup \{\mathrm{\infty }\}}$ defined on then-dimensional unit ball we have:

Intuitively, the blow-up off at 0 "counts for less" whenn is large since the unit ball has "more outside and less inside" in higher dimensions.

Let${\displaystyle 1⩽p⩽\mathrm{\infty }.}$ If a function is in${\displaystyle W^{1,p}(\mathrm{\Omega }),}$ then, possibly after modifying the function on a set of measure zero, the restriction toalmost every line parallel to the coordinate directions in${\displaystyle {\mathbb{R}}^n}$ isabsolutely continuous; what's more, the classical derivative along the lines that are parallel to the coordinate directions are in${\displaystyle L^p(\mathrm{\Omega }).}$ Conversely, if the restriction of${\displaystyle f}$ to almost every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient${\displaystyle \mathrm{\nabla }f}$ existsalmost everywhere, and${\displaystyle f}$ is in${\displaystyle W^{1,p}(\mathrm{\Omega })}$ provided${\displaystyle f,|\mathrm{\nabla }f|\in L^p(\mathrm{\Omega }).}$ In particular, in this case the weak partial derivatives of${\displaystyle f}$ and pointwise partial derivatives of${\displaystyle f}$ agree almost everywhere. The ACL characterization of the Sobolev spaces was established byOtto M. Nikodym (1933); see (Maz'ya 2011, §1.1.3).

A stronger result holds when${\displaystyle p>n.}$ A function in${\displaystyle W^{1,p}(\mathrm{\Omega })}$ is, after modifying on a set of measure zero,Hölder continuous of exponent${\displaystyle \gamma =1-\frac{n}{p},}$ byMorrey's inequality. In particular, if${\displaystyle p=\mathrm{\infty }}$ and${\displaystyle \mathrm{\Omega }}$ has Lipschitz boundary, then the function isLipschitz continuous.

The Sobolev space${\displaystyle W^{1,2}(\mathrm{\Omega })}$ is also denoted by${\displaystyle H^1(\mathrm{\Omega }).}$ It is a Hilbert space, with an important subspace${\displaystyle H_0^1(\mathrm{\Omega })}$ defined to be the closure of the infinitely differentiable functions compactly supported in${\displaystyle \mathrm{\Omega }}$ in${\displaystyle H^1(\mathrm{\Omega }).}$ The Sobolev norm defined above reduces here to

${\displaystyle \Vert f{\Vert }_{H^1}={\left({\int }_{\mathrm{\Omega }}|f{|}^2+|\mathrm{\nabla }f{|}^2\right)}^{\frac{1}{2}}.}$

When${\displaystyle \mathrm{\Omega }}$ has a regular boundary,${\displaystyle H_0^1(\mathrm{\Omega })}$ can be described as the space of functions in${\displaystyle H^1(\mathrm{\Omega })}$ that vanish at the boundary, in the sense of traces (see below). When${\displaystyle n=1,}$ if${\displaystyle \mathrm{\Omega }=(a,b)}$ is a bounded interval, then${\displaystyle H_0^1(a,b)}$ consists of continuous functions on${\displaystyle [a,b]}$ of the form

${\displaystyle f(x)={\int }_a^x{f}^{\prime }(t)\mathrm{d}t,x\in [a,b]}$

where the generalized derivative${\displaystyle f^{\prime }}$ is in${\displaystyle L^2(a,b)}$ and has 0 integral, so that${\displaystyle f(b)=f(a)=0.}$

When${\displaystyle \mathrm{\Omega }}$ is bounded, thePoincaré inequality states that there is a constant${\displaystyle C=C(\mathrm{\Omega })}$ such that:

${\displaystyle {\int }_{\mathrm{\Omega }}|f{|}^2⩽C^2{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }f{|}^2,f\in H_0^1(\mathrm{\Omega }).}$

When${\displaystyle \mathrm{\Omega }}$ is bounded, the injection from${\displaystyle H_0^1(\mathrm{\Omega })}$ to${\displaystyle L^2(\mathrm{\Omega }),}$ iscompact. This fact plays a role in the study of theDirichlet problem, and in the fact that there exists anorthonormal basis of${\displaystyle L^2(\mathrm{\Omega })}$ consisting of eigenvectors of theLaplace operator (withDirichlet boundary condition).

Sobolev spaces are often considered when investigating partial differential equations. It is essential to consider boundary values of Sobolev functions. If${\displaystyle u\in C(\mathrm{\Omega })}$, those boundary values are described by the restriction${\displaystyle u{|}_{\mathrm{\partial }\mathrm{\Omega }}.}$ However, it is not clear how to describe values at the boundary for${\displaystyle u\in W^{k,p}(\mathrm{\Omega }),}$ as then-dimensional measure of the boundary is zero. The following theorem^[2] resolves the problem:

Tu is called the trace ofu. Roughly speaking, this theorem extends the restriction operator to the Sobolev space${\displaystyle W^{1,p}(\mathrm{\Omega })}$ for well-behaved Ω. Note that thetrace operatorT is in general not surjective, but for 1 <p < ∞ it maps continuously onto the Sobolev–Slobodeckij space${\displaystyle W^{1-\frac{1}{p},p}(\mathrm{\partial }\mathrm{\Omega }).}$

Intuitively, taking the trace costs 1/p of a derivative. The functionsu inW^1,p(Ω) with zero trace, i.e.Tu = 0, can be characterized by the equality

${\displaystyle W_0^{1,p}(\mathrm{\Omega })=\left\{u\in W^{1,p}(\mathrm{\Omega }):Tu=0\right\},}$

where

${\displaystyle W_0^{1,p}(\mathrm{\Omega }):=\left\{u\in W^{1,p}(\mathrm{\Omega }):\mathrm{\exists }\{u_m{\}}_{m=1}^{\mathrm{\infty }}\subset C_c^{\mathrm{\infty }}(\mathrm{\Omega }),\text{such that}{u}_m\to u\text{in}{W}^{1,p}(\mathrm{\Omega })\right\}.}$

In other words, for Ω bounded with Lipschitz boundary, trace-zero functions in${\displaystyle W^{1,p}(\mathrm{\Omega })}$ can be approximated by smooth functions with compact support.

For a natural numberk and1 <p < ∞ one can show (by usingFourier multipliers^[3][4]) that the space${\displaystyle W^{k,p}({\mathbb{R}}^n)}$ can equivalently be defined as

${\displaystyle W^{k,p}({\mathbb{R}}^n)=H^{k,p}({\mathbb{R}}^n):=\{f\in L^p({\mathbb{R}}^n):{\mathcal{F}}^{-1}[(1+|\xi {|}^2{)}^{\frac{k}{2}}\mathcal{F}f]\in L^p({\mathbb{R}}^n)\},}$

with the norm

${\displaystyle \Vert f{\Vert }_{H^{k,p}({\mathbb{R}}^n)}:={\Vert {\mathcal{F}}^{-1}[(1+|\xi {|}^2{)}^{\frac{k}{2}}\mathcal{F}f]\Vert }_{L^p({\mathbb{R}}^n)}.}$

This motivates Sobolev spaces with non-integer order since in the above definition we can replacek by any real numbers. The resulting spaces

${\displaystyle H^{s,p}({\mathbb{R}}^n):=\left\{f\in {\mathcal{S}}^{\prime }({\mathbb{R}}^n):{\mathcal{F}}^{-1}\left[(1+|\xi {|}^2{)}^{\frac{s}{2}}\mathcal{F}f\right]\in L^p({\mathbb{R}}^n)\right\}}$

are called Bessel potential spaces^[5] (named afterFriedrich Bessel). They are Banach spaces in general and Hilbert spaces in the special casep = 2.

For${\displaystyle s\ge 0,H^{s,p}(\mathrm{\Omega })}$ is the set of restrictions of functions from${\displaystyle H^{s,p}({\mathbb{R}}^n)}$ to Ω equipped with the norm

${\displaystyle \Vert f{\Vert }_{H^{s,p}(\mathrm{\Omega })}:=inf\left\{\Vert g{\Vert }_{H^{s,p}({\mathbb{R}}^n)}:g\in H^{s,p}({\mathbb{R}}^n),g{|}_{\mathrm{\Omega }}=f\right\}.}$

Again,H^s,p(Ω) is a Banach space and in the casep = 2 a Hilbert space.

Using extension theorems for Sobolev spaces, it can be shown that alsoW^k,p(Ω) =H^k,p(Ω) holds in the sense of equivalent norms, if Ω is domain with uniformC^k-boundary,k a natural number and1 <p < ∞. By theembeddings

${\displaystyle H^{k+1,p}({\mathbb{R}}^n)\hookrightarrow H^{s^{\prime },p}({\mathbb{R}}^n)\hookrightarrow H^{s,p}({\mathbb{R}}^n)\hookrightarrow H^{k,p}({\mathbb{R}}^n),k⩽s⩽s^{\prime }⩽k+1}$

the Bessel potential spaces${\displaystyle H^{s,p}({\mathbb{R}}^n)}$ form a continuous scale between the Sobolev spaces${\displaystyle W^{k,p}({\mathbb{R}}^n).}$ From an abstract point of view, the Bessel potential spaces occur as complexinterpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms it holds that

${\displaystyle {\left[W^{k,p}({\mathbb{R}}^n),W^{k+1,p}({\mathbb{R}}^n)\right]}_{\theta }=H^{s,p}({\mathbb{R}}^n),}$

where:

Another approach to define fractional order Sobolev spaces arises from the idea to generalize theHölder condition to theL^p-setting.^[6] For and${\displaystyle f\in L^p(\mathrm{\Omega }),}$ theSlobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by

${\displaystyle [f{]}_{\theta ,p,\mathrm{\Omega }}:={\left({\int }_{\mathrm{\Omega }}{\int }_{\mathrm{\Omega }}\frac{|f(x)-f(y){|}^p}{|x-y{|}^{\theta p+n}}dxdy\right)}^{\frac{1}{p}}.}$

Lets > 0 be not an integer and set${\displaystyle \theta =s-\lfloor s\rfloor \in (0,1)}$. Using the same idea as for theHölder spaces, theSobolev–Slobodeckij space^[7]${\displaystyle W^{s,p}(\mathrm{\Omega })}$ is defined as

It is a Banach space for the norm

${\displaystyle \Vert f{\Vert }_{W^{s,p}(\mathrm{\Omega })}:=\Vert f{\Vert }_{W^{\lfloor s\rfloor ,p}(\mathrm{\Omega })}+\underset{|\alpha |=\lfloor s\rfloor }{sup}[D^{\alpha }f{]}_{\theta ,p,\mathrm{\Omega }}.}$

If${\displaystyle \mathrm{\Omega }}$ 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 orembeddings

${\displaystyle W^{k+1,p}(\mathrm{\Omega })\hookrightarrow W^{s^{\prime },p}(\mathrm{\Omega })\hookrightarrow W^{s,p}(\mathrm{\Omega })\hookrightarrow W^{k,p}(\mathrm{\Omega }),k⩽s⩽s^{\prime }⩽k+1.}$

There are examples of irregular Ω such that${\displaystyle W^{1,p}(\mathrm{\Omega })}$ is not even a vector subspace of${\displaystyle W^{s,p}(\mathrm{\Omega })}$ for 0 <s < 1 (see Example 9.1 of^[8])

From an abstract point of view, the spaces${\displaystyle W^{s,p}(\mathrm{\Omega })}$ coincide with the realinterpolation spaces of Sobolev spaces, i.e. in the sense of equivalent norms the following holds:

${\displaystyle W^{s,p}(\mathrm{\Omega })={\left(W^{k,p}(\mathrm{\Omega }),W^{k+1,p}(\mathrm{\Omega })\right)}_{\theta ,p},k\in \mathbb{N},s\in (k,k+1),\theta =s-\lfloor s\rfloor .}$

Sobolev–Slobodeckij spaces play an important role in the study of traces of Sobolev functions. They are special cases ofBesov spaces.^[4]

The constant arising in the characterization of the fractional Sobolev space${\displaystyle W^{s,p}(\mathrm{\Omega })}$ can be characterized through the Bourgain-Brezis-Mironescu formula:

${\displaystyle \underset{s\nearrow 1}{lim}(1-s){\int }_{\mathrm{\Omega }}{\int }_{\mathrm{\Omega }}\frac{|f(x)-f(y){|}^p}{|x-y{|}^{sp+n}}dxdy=\frac{2{\pi }^{\frac{n-1}{2}}\mathrm{\Gamma }(\frac{p+1}{2})}{p\mathrm{\Gamma }(\frac{p+n}{2})}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }f{|}^p;}$

and the condition

characterizes those functions of${\displaystyle L^p(\mathrm{\Omega })}$ that are in the first-order Sobolev space${\displaystyle W^{1,p}(\mathrm{\Omega })}$.^[9]

If${\displaystyle \mathrm{\Omega }}$ is adomain whose boundary is not too poorly behaved (e.g., if its boundary is a manifold, or satisfies the more permissive "cone condition") then there is an operatorA mapping functions of${\displaystyle \mathrm{\Omega }}$ to functions of${\displaystyle {\mathbb{R}}^n}$ such that:

1. Au(x) =u(x) for almost everyx in${\displaystyle \mathrm{\Omega }}$ and
2. ${\displaystyle A:W^{k,p}(\mathrm{\Omega })\to W^{k,p}({\mathbb{R}}^n)}$ is continuous for any 1 ≤p ≤ ∞ and integerk.

We will call such an operatorA an extension operator for${\displaystyle \mathrm{\Omega }.}$

Extension operators are the most natural way to define${\displaystyle H^s(\mathrm{\Omega })}$ for non-integers (we cannot work directly on${\displaystyle \mathrm{\Omega }}$ since takingFourier transform is a global operation). We define${\displaystyle H^s(\mathrm{\Omega })}$ by saying that${\displaystyle u\in H^s(\mathrm{\Omega })}$ if and only if${\displaystyle Au\in H^s({\mathbb{R}}^n).}$ Equivalently, complex interpolation yields the same${\displaystyle H^s(\mathrm{\Omega })}$ spaces so long as${\displaystyle \mathrm{\Omega }}$ has an extension operator. If${\displaystyle \mathrm{\Omega }}$ does not have an extension operator, complex interpolation is the only way to obtain the${\displaystyle H^s(\mathrm{\Omega })}$ spaces.

As a result, the interpolation inequality still holds.

Likeabove, we define${\displaystyle H_0^s(\mathrm{\Omega })}$ to be the closure in${\displaystyle H^s(\mathrm{\Omega })}$ of the space${\displaystyle C_c^{\mathrm{\infty }}(\mathrm{\Omega })}$ of infinitely differentiable compactly supported functions. Given the definition of a trace, above, we may state the following

If${\displaystyle u\in H_0^s(\mathrm{\Omega })}$ we may define itsextension by zero${\displaystyle \tilde{u}\in L^2({\mathbb{R}}^n)}$ in the natural way, namely

Forf ∈L^p(Ω) its extension by zero,

is an element of${\displaystyle L^p({\mathbb{R}}^n).}$ Furthermore,

${\displaystyle \Vert Ef{\Vert }_{L^p({\mathbb{R}}^n)}=\Vert f{\Vert }_{L^p(\mathrm{\Omega })}.}$

In the case of the Sobolev spaceW^1,p(Ω) for1 ≤ p ≤ ∞, extending a functionu by zero will not necessarily yield an element of${\displaystyle W^{1,p}({\mathbb{R}}^n).}$ But if Ω is bounded with Lipschitz boundary (e.g. ∂Ω isC¹), 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^[2]

${\displaystyle E:W^{1,p}(\mathrm{\Omega })\to W^{1,p}({\mathbb{R}}^n),}$

such that for each${\displaystyle u\in W^{1,p}(\mathrm{\Omega }):Eu=u}$ a.e. on Ω,Eu has compact support within O, and there exists a constantC depending only onp, Ω, O and the dimensionn, such that

${\displaystyle \Vert Eu{\Vert }_{W^{1,p}({\mathbb{R}}^n)}⩽C\Vert u{\Vert }_{W^{1,p}(\mathrm{\Omega })}.}$

We call${\displaystyle Eu}$ an extension of${\displaystyle u}$ to${\displaystyle {\mathbb{R}}^n.}$

It is a natural question to ask if a Sobolev function is continuous or even continuously differentiable. Roughly speaking, sufficiently many weak derivatives (i.e. largek) result in a classical derivative. This idea is generalized and made precise in theSobolev embedding theorem.

Write${\displaystyle W^{k,p}}$ for the Sobolev space of some compactRiemannian manifold of dimensionn. Herek can be anyreal number, and 1 ≤ p ≤ ∞. (Forp = ∞ the Sobolev space${\displaystyle W^{k,\mathrm{\infty }}}$ is defined to be theHölder spaceC^n,α wherek = n + α and 0 < α ≤ 1.) The Sobolev embedding theorem states that if${\displaystyle k⩾m}$ and${\displaystyle k-\frac{n}{p}⩾m-\frac{n}{q}}$ then

${\displaystyle W^{k,p}\subseteq W^{m,q}}$

and the embedding is continuous. Moreover, if${\displaystyle k>m}$ and${\displaystyle k-\frac{n}{p}>m-\frac{n}{q}}$ then the embedding is completely continuous (this is sometimes called Kondrachov's theorem or theRellich–Kondrachov theorem). Functions in${\displaystyle W^{m,\mathrm{\infty }}}$ have all derivatives of order less thanm continuous, so in particular this gives conditions on Sobolev spaces for various derivatives to be continuous. Informally these embeddings say that to convert anL^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${\displaystyle {\mathbb{R}}^n}$ (Stein 1970). Sobolev embeddings on${\displaystyle {\mathbb{R}}^n}$ that are not compact often have a related, but weaker, property ofcocompactness.

• Sobolev mapping
• Souček space
• Besov space
• Triebel–Lizorkin space

• Adams, Robert A.; Fournier, John (2003) [1975].Sobolev Spaces. Pure and Applied Mathematics. Vol. 140 (2nd ed.). Boston, MA:Academic Press.ISBN 978-0-12-044143-3..
• Aubin, Thierry (1982),Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Berlin, New York:Springer-Verlag,doi:10.1007/978-1-4612-5734-9,ISBN 978-0-387-90704-8,MR 0681859.
• Bergh, Jöran; Löfström, Jörgen (1976),Interpolation Spaces, An Introduction, Grundlehren der Mathematischen Wissenschaften, vol. 223, Springer-Verlag, pp. X + 207,ISBN 978-7-5062-6011-4,MR 0482275,Zbl 0344.46071
• Evans, Lawrence C. (2010) [1998].Partial Differential Equations.Graduate Studies in Mathematics. Vol. 19 (2nd ed.). American Mathematical Society. p. 749.ISBN 978-0-8218-4974-3.
• Leoni, Giovanni (2009).A First Course in Sobolev Spaces.Graduate Studies in Mathematics. Vol. 105. American Mathematical Society. pp. xvi+607.ISBN 978-0-8218-4768-8.MR 2527916.Zbl 1180.46001.
• Maz'ja, Vladimir G. (1985),Sobolev Spaces, Springer Series in Soviet Mathematics, Berlin–Heidelberg–New York:Springer-Verlag, pp. xix+486,doi:10.1007/978-3-662-09922-3,ISBN 0-387-13589-8,MR 0817985,Zbl 0692.46023
• Maz'ya, Vladimir G.; Poborchi, Sergei V. (1997),Differentiable Functions on Bad Domains, Singapore–New Jersey–London–Hong Kong:World Scientific, pp. xx+481,ISBN 981-02-2767-1,MR 1643072,Zbl 0918.46033.
• Maz'ya, Vladimir G. (2011) [1985],Sobolev Spaces. With Applications to Elliptic Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften, vol. 342 (2nd revised and augmented ed.), Berlin–Heidelberg–New York:Springer Verlag, pp. xxviii+866,doi:10.1007/978-3-642-15564-2,ISBN 978-3-642-15563-5,MR 2777530,Zbl 1217.46002.
• Lunardi, Alessandra (1995),Analytic semigroups and optimal regularity in parabolic problems, Basel:Birkhäuser Verlag.
• Nikodym, Otto (1933),"Sur une classe de fonctions considérée dans l'étude du problème de Dirichlet",Fund. Math.,21:129–150,doi:10.4064/fm-21-1-129-150.
• 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.
• Sobolev, S. L. (1963), "On a theorem of functional analysis",Eleven Papers on Analysis, American Mathematical Society Translations: Series 2, vol. 34, pp. 39–68,doi:10.1090/trans2/034/02,ISBN 9780821817346{{citation}}:ISBN / Date incompatibility (help); translation of Mat. Sb., 4 (1938) pp. 471–497.
• Sobolev, S.L. (1963),Some applications of functional analysis in mathematical physics, Amer. Math. Soc..
• Stein, E (1970),Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press,ISBN 0-691-08079-8.
• Triebel, H. (1995),Interpolation Theory, Function Spaces, Differential Operators, Heidelberg: Johann Ambrosius Barth.
• Ziemer, William P. (1989),Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Berlin, New York:Springer-Verlag,doi:10.1007/978-1-4612-1015-3,hdl:10338.dmlcz/143849,ISBN 978-0-387-97017-2,MR 1014685.

• Eleonora Di Nezza, Giampiero Palatucci, Enrico Valdinoci (2011). "Hitchhiker's guide to the fractional Sobolev spaces".

• Sobolev spaces
• Fourier analysis
• Fractional calculus
• Function spaces

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