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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 and 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. Intuitively, a Sobolev space is a 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 regularity of a function.

Sobolev spaces are named after the Russian mathematician 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 functions and with the derivatives understood in the classical sense.

Motivation

In this section and throughout the article Ω{displaystyle Omega } is an open subset of Rn.{displaystyle mathbb {R} ^{n}.}

There are many criteria for smoothness of mathematical functions. The most basic criterion may be that of 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 C¹ — see Differentiability class). 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.

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 L²-norm. It is therefore important to develop a tool for differentiating Lebesgue space functions.

The integration by parts formula yields that for every u ∈ C^k(Ω), where k is a natural number, and for all infinitely differentiable functions with compact support φ∈Cc∞(Ω),{displaystyle varphi in C_{c}^{infty }(Omega ),}

∫ΩuDαφdx=(−1)|α|∫ΩφDαudx,{displaystyle int _{Omega }uD^{alpha }varphi ;dx=(-1)^{|alpha |}int _{Omega }varphi D^{alpha }u;dx,}

where α a multi-index of order |α| = k and we are using the notation:

Dαf=∂|α|f∂x1α1…∂xnαn.{displaystyle D^{alpha }f={frac {partial ^{|alpha |}f}{partial x_{1}^{alpha _{1}}dots partial x_{n}^{alpha _{n}}}}.}

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

∫ΩuDαφdx=(−1)|α|∫Ωφvdx,φ∈Cc∞(Ω),{displaystyle int _{Omega }uD^{alpha }varphi ;dx=(-1)^{|alpha |}int _{Omega }varphi v;dx,qquad varphi in C_{c}^{infty }(Omega ),}

we call v the 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, and thus it is uniquely determined as an element of a Lebesgue space. On the other hand, if u ∈ C^k(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^αu := v.

For example, the function

u(x)={1+x−1<x<010x=01−x0<x<10else{displaystyle u(x)={begin{cases}1+x&-1<x<0\10&x=0\1-x&0<x<1\0&{text{else}}end{cases}}}

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

v(x)={1−1<x<0−10<x<10else{displaystyle v(x)={begin{cases}1&-1<x<0\-1&0<x<1\0&{text{else}}end{cases}}}

satisfies the definition for being the weak derivative of u(x),{displaystyle u(x),} which then qualifies as being in the Sobolev space W1,p{displaystyle W^{1,p}} (for any allowed p, see definition below).

The Sobolev spaces Wk,p(Ω){displaystyle W^{k,p}(Omega )} combine the concepts of weak differentiability and Lebesgue norms.

Sobolev spaces with integer k

One-dimensional case

In the one-dimensional case the Sobolev space Wk,p(R){displaystyle W^{k,p}(mathbb {R} )} for 1⩽p⩽∞{displaystyle 1leqslant pleqslant infty } is defined as the subset of functions f{displaystyle f} in Lp(R){displaystyle L^{p}(mathbb {R} )} such that f{displaystyle f} and its weak derivatives up to order k{displaystyle k} have a finite L^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 f(k−1),{displaystyle f^{(k-1)},} the (k−1){displaystyle (k-1)}–th derivative of the function f,{displaystyle f,} 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).

With this definition, the Sobolev spaces admit a natural norm,

‖f‖k,p=(∑i=0k‖f(i)‖pp)1p=(∑i=0k∫|f(i)(t)|pdt)1p.{displaystyle |f|_{k,p}=left(sum _{i=0}^{k}left|f^{(i)}right|_{p}^{p}right)^{frac {1}{p}}=left(sum _{i=0}^{k}int left|f^{(i)}(t)right|^{p},dtright)^{frac {1}{p}}.}

Equipped with the norm ‖⋅‖k,p,Wk,p{displaystyle |cdot |_{k,p},W^{k,p}} becomes 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

‖f(k)‖p+‖f‖p{displaystyle left|f^{(k)}right|_{p}+|f|_{p}}

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

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, since the space is a Hilbert space:

Hk=Wk,2.{displaystyle H^{k}=W^{k,2}.}

The space Hk{displaystyle H^{k}} can be defined naturally in terms of Fourier series whose coefficients decay sufficiently rapidly, namely,

Hk(T)={f∈L2(T):∑n=−∞∞(1+n2+n4+⋯+n2k)|f^(n)|2<∞}{displaystyle H^{k}(mathbb {T} )=left{fin L^{2}(mathbb {T} ):sum _{n=-infty }^{infty }left(1+n^{2}+n^{4}+dots +n^{2k}right)left|{widehat {f}}(n)right|^{2}<infty right}}

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

‖f‖k,22=∑n=−∞∞(1+|n|2)k|f^(n)|2.{displaystyle |f|_{k,2}^{2}=sum _{n=-infty }^{infty }left(1+|n|^{2}right)^{k}left|{widehat {f}}(n)right|^{2}.}

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 Hk{displaystyle H^{k}} admits an inner product, like the space H0=L2.{displaystyle H^{0}=L^{2}.} In fact, the Hk{displaystyle H^{k}} inner product is defined in terms of the L2{displaystyle L^{2}} inner product:

⟨u,v⟩Hk=∑i=0k⟨Diu,Div⟩L2.{displaystyle langle u,vrangle _{H^{k}}=sum _{i=0}^{k}leftlangle D^{i}u,D^{i}vrightrangle _{L^{2}}.}

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

Other examples

Some other Sobolev spaces permit a simpler description. For example, W1,1(0,1){displaystyle W^{1,1}(0,1)} is the space of absolutely continuous functions on (0, 1) (or rather, equivalence classes of functions that are equal almost everywhere to such), while W1,∞(I){displaystyle W^{1,infty }(I)} is the space of Lipschitz functions on I, for every interval I. All spaces Wk,∞{displaystyle W^{k,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 p<∞.{displaystyle p<infty .} (E.g., functions behaving like |x|^−1/3 at the origin are in L2,{displaystyle L^{2},} but the product of two such functions is not in L2{displaystyle L^{2}}).

Multidimensional case

The transition to multiple dimensions brings more difficulties, starting from the very definition. The requirement that f(k−1){displaystyle f^{(k-1)}} be the integral of f(k){displaystyle 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 k∈N,1⩽p⩽∞.{displaystyle kin mathbb {N} ,1leqslant pleqslant infty .} The Sobolev space Wk,p(Ω){displaystyle W^{k,p}(Omega )} is defined to be the set of all functions f{displaystyle f} on Ω{displaystyle Omega } such that for every multi-index α{displaystyle alpha } with |α|⩽k,{displaystyle |alpha |leqslant k,} the mixed partial derivative

f(α)=∂|α|f∂x1α1…∂xnαn{displaystyle f^{(alpha )}={frac {partial ^{|alpha |}f}{partial x_{1}^{alpha _{1}}dots partial x_{n}^{alpha _{n}}}}}

exists in the weak sense and is in Lp(Ω),{displaystyle L^{p}(Omega ),} i.e.

‖f(α)‖Lp<∞.{displaystyle left|f^{(alpha )}right|_{L^{p}}<infty .}

That is, the Sobolev space Wk,p(Ω){displaystyle W^{k,p}(Omega )} is defined as

Wk,p(Ω)={u∈Lp(Ω):Dαu∈Lp(Ω)∀|α|⩽k}.{displaystyle W^{k,p}(Omega )=left{uin L^{p}(Omega ):D^{alpha }uin L^{p}(Omega ),,forall |alpha |leqslant kright}.}

The natural number k{displaystyle k} is called the order of the Sobolev space Wk,p(Ω).{displaystyle W^{k,p}(Omega ).}

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

‖u‖Wk,p(Ω):={(∑|α|⩽k‖Dαu‖Lp(Ω)p)1p1⩽p<∞;max|α|⩽k‖Dαu‖L∞(Ω)p=∞;{displaystyle |u|_{W^{k,p}(Omega )}:={begin{cases}left(sum _{|alpha |leqslant k}left|D^{alpha }uright|_{L^{p}(Omega )}^{p}right)^{frac {1}{p}}&1leqslant p<infty ;\max _{|alpha |leqslant k}left|D^{alpha }uright|_{L^{infty }(Omega )}&p=infty ;end{cases}}}

and

‖u‖Wk,p(Ω)′:={∑|α|⩽k‖Dαu‖Lp(Ω)1⩽p<∞;∑|α|⩽k‖Dαu‖L∞(Ω)p=∞.{displaystyle |u|'_{W^{k,p}(Omega )}:={begin{cases}sum _{|alpha |leqslant k}left|D^{alpha }uright|_{L^{p}(Omega )}&1leqslant p<infty ;\sum _{|alpha |leqslant k}left|D^{alpha }uright|_{L^{infty }(Omega )}&p=infty .end{cases}}}

With respect to either of these norms, Wk,p(Ω){displaystyle W^{k,p}(Omega )} is a Banach space. For p<∞,Wk,p(Ω){displaystyle p<infty ,W^{k,p}(Omega )} is also a separable space. It is conventional to denote Wk,2(Ω){displaystyle W^{k,2}(Omega )} by Hk(Ω){displaystyle H^{k}(Omega )} for it is a Hilbert space with the norm ‖⋅‖Wk,2(Ω){displaystyle |cdot |_{W^{k,2}(Omega )}}.^[1]

Approximation by smooth functions

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

‖um−u‖Wk,p(Ω)→0.{displaystyle left|u_{m}-uright|_{W^{k,p}(Omega )}to 0.}

If Ω{displaystyle Omega } has Lipschitz boundary, we may even assume that the um{displaystyle u_{m}} are the restriction of smooth functions with compact support on all of Rn.{displaystyle mathbb {R} ^{n}.}^[2]

Examples

In higher dimensions, it is no longer true that, for example, W1,1{displaystyle W^{1,1}} contains only continuous functions. For example, |x|−1∈W1,1(B3){displaystyle |x|^{-1}in W^{1,1}(mathbb {B} ^{3})} where B3{displaystyle mathbb {B} ^{3}} is the unit ball in three dimensions. For k > n/p the space Wk,p(Ω){displaystyle W^{k,p}(Omega )} 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 for the function f:Bn→R∪{∞}{displaystyle f:mathbb {B} ^{n}to mathbb {R} cup {infty }} defined on the n-dimensional ball we have:

f(x)=|x|−α∈Wk,p(Bn)⟺α<np−k.{displaystyle f(x)=|x|^{-alpha }in W^{k,p}(mathbb {B} ^{n})Longleftrightarrow alpha <{tfrac {n}{p}}-k.}

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

Absolutely continuous on lines (ACL) characterization of Sobolev functions

Let 1⩽p⩽∞.{displaystyle 1leqslant pleqslant infty .} If a function is in W1,p(Ω),{displaystyle W^{1,p}(Omega ),} then, possibly after modifying the function on a set of measure zero, the restriction to almost every line parallel to the coordinate directions in Rn{displaystyle mathbb {R} ^{n}} is absolutely continuous; what's more, the classical derivative along the lines that are parallel to the coordinate directions are in Lp(Ω).{displaystyle L^{p}(Omega ).} Conversely, if the restriction of f{displaystyle f} to almost every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient ∇f{displaystyle nabla f} exists almost everywhere, and f{displaystyle f} is in W1,p(Ω){displaystyle W^{1,p}(Omega )} provided f,|∇f|∈Lp(Ω).{displaystyle f,|nabla f|in L^{p}(Omega ).} In particular, in this case the weak partial derivatives of f{displaystyle f} and pointwise partial derivatives of f{displaystyle f} agree almost everywhere. The ACL characterization of the Sobolev spaces was established by Otto M. Nikodym (1933); see (Maz'ya 1985, §1.1.3).

A stronger result holds when p>n.{displaystyle p>n.} A function in W1,p(Ω){displaystyle W^{1,p}(Omega )} is, after modifying on a set of measure zero, Hölder continuous of exponent γ=1−np,{displaystyle gamma =1-{tfrac {n}{p}},} by Morrey's inequality. In particular, if p=∞,{displaystyle p=infty ,} then the function is Lipschitz continuous.

Functions vanishing at the boundary

The Sobolev space W1,2(Ω){displaystyle W^{1,2}(Omega )} is also denoted by H1(Ω).{displaystyle H^{1}(Omega ).} It is a Hilbert space, with an important subspace H01(Ω){displaystyle H_{0}^{1}(Omega )} defined to be the closure of the infinitely differentiable functions compactly supported in Ω{displaystyle Omega } in H1(Ω).{displaystyle H^{1}(Omega ).} The Sobolev norm defined above reduces here to

‖f‖H1=(∫Ω(|f|2+|∇f|2))12.{displaystyle |f|_{H^{1}}=left(int _{Omega }left(|f|^{2}+|nabla f|^{2}right)right)^{frac {1}{2}}.}

When Ω{displaystyle Omega } has a regular boundary, H01(Ω){displaystyle H_{0}^{1}(Omega )} can be described as the space of functions in H1(Ω){displaystyle H^{1}(Omega )} that vanish at the boundary, in the sense of traces (see below). When n=1,{displaystyle n=1,} if Ω=(a,b){displaystyle Omega =(a,b)} is a bounded interval, then H01(a,b){displaystyle H_{0}^{1}(a,b)} consists of continuous functions on [a,b]{displaystyle [a,b]} of the form

f(x)=∫axf′(t)dt,x∈[a,b]{displaystyle f(x)=int _{a}^{x}f'(t),mathrm {d} t,qquad xin [a,b]}

where the generalized derivative f′{displaystyle f'} is in L2(a,b){displaystyle L^{2}(a,b)} and has 0 integral, so that f(b)=f(a)=0.{displaystyle f(b)=f(a)=0.}

When Ω{displaystyle Omega } is bounded, the Poincaré inequality states that there is a constant C=C(Ω){displaystyle C=C(Omega )} such that:

∫Ω|f|2⩽C2∫Ω|∇f|2,f∈H01(Ω).{displaystyle int _{Omega }|f|^{2}leqslant C^{2}int _{Omega }|nabla f|^{2},qquad fin H_{0}^{1}(Omega ).}

When Ω{displaystyle Omega } is bounded, the injection from H01(Ω){displaystyle H_{0}^{1}(Omega )} to L2(Ω),{displaystyle L^{2}(Omega ),} is compact. This fact plays a role in the study of the Dirichlet problem, and in the fact that there exists an orthonormal basis of L2(Ω){displaystyle L^{2}(Omega )} consisting of eigenvectors of the Laplace operator (with Dirichlet boundary condition).

Traces

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 u|∂Ω{displaystyle u|_{partial Omega }}. However, it is not clear how to describe values at the boundary for u ∈ W^k,p(Ω), as the n-dimensional measure of the boundary is zero. The following theorem^[2] resolves the problem:

Trace Theorem. Assume Ω is bounded with Lipschitz boundary. Then there exists a bounded linear operator T:W1,p(Ω)→Lp(∂Ω){displaystyle T:W^{1,p}(Omega )to L^{p}(partial Omega )} such that

Tu=u|∂Ωu∈W1,p(Ω)∩C(Ω¯)‖Tu‖Lp(∂Ω)⩽c(p,Ω)‖u‖W1,p(Ω)u∈W1,p(Ω).{displaystyle {begin{aligned}Tu&=u|_{partial Omega }&&uin W^{1,p}(Omega )cap C({overline {Omega }})\|Tu|_{L^{p}(partial Omega )}&leqslant c(p,Omega )|u|_{W^{1,p}(Omega )}&&uin W^{1,p}(Omega ).end{aligned}}}

Tu is called the trace of u. Roughly speaking, this theorem extends the restriction operator to the Sobolev space W1,p(Ω){displaystyle W^{1,p}(Omega )} 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 W1−1p,p(∂Ω).{displaystyle W^{1-{frac {1}{p}},p}(partial Omega ).}

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

W01,p(Ω)={u∈W1,p(Ω):Tu=0},{displaystyle W_{0}^{1,p}(Omega )=left{uin W^{1,p}(Omega ):Tu=0right},}

where

W01,p(Ω):={u∈W1,p(Ω):∃{um}m=1∞⊂Cc∞(Ω), such that um→u in W1,p(Ω)}.{displaystyle W_{0}^{1,p}(Omega ):=left{uin W^{1,p}(Omega ):exists {u_{m}}_{m=1}^{infty }subset C_{c}^{infty }(Omega ), {text{such that}} u_{m}to u {textrm {in}} W^{1,p}(Omega )right}.}

In other words, for Ω bounded with Lipschitz boundary, trace-zero functions in W1,p(Ω){displaystyle W^{1,p}(Omega )} can be approximated by smooth functions with compact support.

Sobolev spaces with non-integer k

Bessel potential spaces

For a natural number k and 1 < p < ∞ one can show (by using Fourier multipliers^[3][4]) that the space Wk,p(Rn){displaystyle W^{k,p}(mathbb {R} ^{n})} can equivalently be defined as

Wk,p(Rn)=Hk,p(Rn):={f∈Lp(Rn):F−1[(1+|ξ|2)k2Ff]∈Lp(Rn)},{displaystyle W^{k,p}(mathbb {R} ^{n})=H^{k,p}(mathbb {R} ^{n}):=left{fin L^{p}(mathbb {R} ^{n}):{mathcal {F}}^{-1}left[(1+|xi |^{2})^{frac {k}{2}}{mathcal {F}}fright]in L^{p}(mathbb {R} ^{n})right},}

with the norm

‖f‖Hk,p(Rn):=‖F−1[(1+|ξ|2)k2Ff]‖Lp(Rn).{displaystyle |f|_{H^{k,p}(mathbb {R} ^{n})}:=left|{mathcal {F}}^{-1}left[left(1+|xi |^{2}right)^{frac {k}{2}}{mathcal {F}}fright]right|_{L^{p}(mathbb {R} ^{n})}.}

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

Hs,p(Rn):={f∈Lp(Rn):F−1[(1+|ξ|2)s2Ff]∈Lp(Rn)}{displaystyle H^{s,p}(mathbb {R} ^{n}):=left{fin L^{p}(mathbb {R} ^{n}):{mathcal {F}}^{-1}left[left(1+|xi |^{2}right)^{frac {s}{2}}{mathcal {F}}fright]in L^{p}(mathbb {R} ^{n})right}}

are called Bessel potential spaces^[5] (named after Friedrich Bessel). They are Banach spaces in general and Hilbert spaces in the special case p = 2.

Hs,p(Ω){displaystyle H^{s,p}(Omega )} is the set of restrictions of functions from Hs,p(Rn){displaystyle H^{s,p}(mathbb {R} ^{n})} to Ω equipped with the norm

‖f‖Hs,p(Ω):=inf{‖g‖Hs,p(Rn):g∈Hs,p(Rn),g|Ω=f}{displaystyle |f|_{H^{s,p}(Omega )}:=inf left{|g|_{H^{s,p}(mathbb {R} ^{n})}:gin H^{s,p}(mathbb {R} ^{n}),g|_{Omega }=fright}}.

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

Using extension theorems for Sobolev spaces, it can be shown that also W^k,p(Ω) = H^k,p(Ω) holds in the sense of equivalent norms, if Ω is domain with uniform C^k-boundary, k a natural number and 1 < p < ∞. By the embeddings

Hk+1,p(Rn)↪Hs′,p(Rn)↪Hs,p(Rn)↪Hk,p(Rn),k⩽s⩽s′⩽k+1{displaystyle 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 kleqslant sleqslant s'leqslant k+1}

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

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

where:

1⩽p⩽∞, 0<θ<1, s=(1−θ)k+θ(k+1)=k+θ.{displaystyle 1leqslant pleqslant infty , 0<theta <1, s=(1-theta )k+theta (k+1)=k+theta .}

Sobolev–Slobodeckij spaces

Another approach to define fractional order Sobolev spaces arises from the idea to generalize the Hölder condition to the L^p-setting.^[6] For 1⩽p<∞,θ∈(0,1){displaystyle 1leqslant p<infty ,theta in (0,1)} and f∈Lp(Ω),{displaystyle fin L^{p}(Omega ),} the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by

[f]θ,p,Ω:=(∫Ω∫Ω|f(x)−f(y)|p|x−y|θp+ndxdy)1p.{displaystyle [f]_{theta ,p,Omega }:=left(int _{Omega }int _{Omega }{frac {|f(x)-f(y)|^{p}}{|x-y|^{theta p+n}}};dx;dyright)^{frac {1}{p}}.}

Let s > 0 be not an integer and set θ=s−⌊s⌋∈(0,1){displaystyle theta =s-lfloor srfloor in (0,1)}. Using the same idea as for the Hölder spaces, the Sobolev–Slobodeckij space^[7]Ws,p(Ω){displaystyle W^{s,p}(Omega )} is defined as

Ws,p(Ω):={f∈W⌊s⌋,p(Ω):sup|α|=⌊s⌋[Dαf]θ,p,Ω<∞}.{displaystyle W^{s,p}(Omega ):=left{fin W^{lfloor srfloor ,p}(Omega ):sup _{|alpha |=lfloor srfloor }[D^{alpha }f]_{theta ,p,Omega }<infty right}.}

It is a Banach space for the norm

‖f‖Ws,p(Ω):=‖f‖W⌊s⌋,p(Ω)+sup|α|=⌊s⌋[Dαf]θ,p,Ω.{displaystyle |f|_{W^{s,p}(Omega )}:=|f|_{W^{lfloor srfloor ,p}(Omega )}+sup _{|alpha |=lfloor srfloor }[D^{alpha }f]_{theta ,p,Omega }.}

If Ω{displaystyle 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 or embeddings

Wk+1,p(Ω)↪Ws′,p(Ω)↪Ws,p(Ω)↪Wk,p(Ω),k⩽s⩽s′⩽k+1.{displaystyle W^{k+1,p}(Omega )hookrightarrow W^{s',p}(Omega )hookrightarrow W^{s,p}(Omega )hookrightarrow W^{k,p}(Omega ),quad kleqslant sleqslant s'leqslant k+1.}

There are examples of irregular Ω such that W1,p(Ω){displaystyle W^{1,p}(Omega )} is not even a vector subspace of Ws,p(Ω){displaystyle W^{s,p}(Omega )} for 0 < s < 1.^{[citation needed]}((Check Example 9.1 in the Hitchhiker guide.))

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

Ws,p(Ω)=(Wk,p(Ω),Wk+1,p(Ω))θ,p,k∈N,s∈(k,k+1),θ=s−⌊s⌋{displaystyle W^{s,p}(Omega )=left(W^{k,p}(Omega ),W^{k+1,p}(Omega )right)_{theta ,p},quad kin mathbb {N} ,sin (k,k+1),theta =s-lfloor srfloor }.

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

Extension operators

If Ω{displaystyle Omega } is a domain 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 operator A mapping functions of Ω{displaystyle Omega } to functions of Rn{displaystyle mathbb {R} ^{n}} such that:

1. 
   Au(x) = u(x) for almost every x in Ω{displaystyle Omega } and


2. 
   A:Wk,p(Ω)→Wk,p(Rn){displaystyle A:W^{k,p}(Omega )to W^{k,p}(mathbb {R} ^{n})} is continuous for any 1 ≤ p ≤ ∞ and integer k.

We will call such an operator A an extension operator for Ω.{displaystyle Omega .}

Case of p = 2

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

As a result, the interpolation inequality still holds.

Extension by zero

Like above, we define H0s(Ω){displaystyle H_{0}^{s}(Omega )} to be the closure in Hs(Ω){displaystyle H^{s}(Omega )} of the space Cc∞(Ω){displaystyle C_{c}^{infty }(Omega )} of infinitely differentiable compactly supported functions. Given the definition of a trace, above, we may state the following

Theorem. Let Ω{displaystyle Omega } be uniformly C^m regular, m ≥ s and let P be the linear map sending u in Hs(Ω){displaystyle H^{s}(Omega )} to

(u,dudn,…,dkudnk)|G{displaystyle left.left(u,{frac {du}{dn}},dots ,{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 H0s{displaystyle H_{0}^{s}} is precisely the kernel of P.

If u∈H0s(Ω){displaystyle uin H_{0}^{s}(Omega )} we may define its extension by zero u~∈L2(Rn){displaystyle {tilde {u}}in L^{2}(mathbb {R} ^{n})} in the natural way, namely

u~(x)={u(x)x∈Ω0else{displaystyle {tilde {u}}(x)={begin{cases}u(x)&xin Omega \0&{text{else}}end{cases}}}

Theorem. Let s>12.{displaystyle s>{tfrac {1}{2}}.} The map u↦u~{displaystyle umapsto {tilde {u}}} is continuous into Hs(Rn){displaystyle H^{s}(mathbb {R} ^{n})} if and only if s is not of the form n+12{displaystyle n+{tfrac {1}{2}}} for n an integer.

For f ∈ L^p(Ω) its extension by zero,

Ef:={fon Ω,0otherwise{displaystyle Ef:={begin{cases}f&{textrm {on}} Omega ,\0&{textrm {otherwise}}end{cases}}}

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

‖Ef‖Lp(Rn)=‖f‖Lp(Ω).{displaystyle |Ef|_{L^{p}(mathbb {R} ^{n})}=|f|_{L^{p}(Omega )}.}

In the case of the Sobolev space W^1,p(Ω) for 1 ≤ p ≤ ∞, extending a function u by zero will not necessarily yield an element of W1,p(Rn).{displaystyle W^{1,p}(mathbb {R} ^{n}).} But if Ω is bounded with Lipschitz boundary (e.g. ∂Ω is C¹), 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]

E:W1,p(Ω)→W1,p(Rn),{displaystyle E:W^{1,p}(Omega )to W^{1,p}(mathbb {R} ^{n}),}

such that for each u∈W1,p(Ω):Eu=u{displaystyle uin W^{1,p}(Omega ):Eu=u} a.e. on Ω, Eu has compact support within O, and there exists a constant C depending only on p, Ω, O and the dimension n, such that

‖Eu‖W1,p(Rn)⩽C‖u‖W1,p(Ω).{displaystyle |Eu|_{W^{1,p}(mathbb {R} ^{n})}leqslant C|u|_{W^{1,p}(Omega )}.}

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

Sobolev embeddings

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 embedding theorem.

Write Wk,p{displaystyle 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 Wk,∞{displaystyle 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⩾m{displaystyle kgeqslant m} and k−np⩾m−nq{displaystyle k-{tfrac {n}{p}}geqslant m-{tfrac {n}{q}}} then

Wk,p⊆Wm,q{displaystyle W^{k,p}subseteq W^{m,q}}

and the embedding is continuous. Moreover, if k>m{displaystyle k>m} and k−np>m−nq{displaystyle k-{tfrac {n}{p}}>m-{tfrac {n}{q}}} then the embedding is completely continuous (this is sometimes called Kondrachov's theorem or the Rellich-Kondrachov theorem). Functions in Wm,∞{displaystyle W^{m,infty }} 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^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{displaystyle mathbb {R} ^{n}} (Stein 1970). Sobolev embeddings on Rn{displaystyle mathbb {R} ^{n}} that are not compact often have a related, but weaker, property of cocompactness.

Notes

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External links

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