Sobolev space




In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-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.




Contents






  • 1 Motivation


  • 2 Sobolev spaces with integer k


    • 2.1 One-dimensional case


      • 2.1.1 The case p = 2


      • 2.1.2 Other examples




    • 2.2 Multidimensional case


      • 2.2.1 Approximation by smooth functions


      • 2.2.2 Examples


      • 2.2.3 Absolutely continuous on lines (ACL) characterization of Sobolev functions


      • 2.2.4 Functions vanishing at the boundary






  • 3 Traces


  • 4 Sobolev spaces with non-integer k


    • 4.1 Bessel potential spaces


    • 4.2 Sobolev–Slobodeckij spaces




  • 5 Extension operators


    • 5.1 Case of p = 2


    • 5.2 Extension by zero




  • 6 Sobolev embeddings


  • 7 Notes


  • 8 References


  • 9 External links





Motivation


In this section and throughout the article Ω{displaystyle Omega }Omega is an open subset of Rn.{displaystyle mathbb {R} ^{n}.}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 uCk(Ω), where k is a natural number, and for all infinitely differentiable functions with compact support φCc∞),{displaystyle varphi in C_{c}^{infty }(Omega ),}{displaystyle varphi in C_{c}^{infty }(Omega ),}


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


f=∂|f∂x1α1…xnαn.{displaystyle D^{alpha }f={frac {partial ^{|alpha |}f}{partial x_{1}^{alpha _{1}}dots partial x_{n}^{alpha _{n}}}}.}{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,φCc∞),{displaystyle int _{Omega }uD^{alpha }varphi ;dx=(-1)^{|alpha |}int _{Omega }varphi v;dx,qquad varphi in C_{c}^{infty }(Omega ),}{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 ∈ Ck(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 Dαu := 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}}}{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}}}{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),}u(x), which then qualifies as being in the Sobolev space W1,p{displaystyle W^{1,p}}W^{1,p} (for any allowed p, see definition below).


The Sobolev spaces Wk,p(Ω){displaystyle W^{k,p}(Omega )}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} )}{displaystyle W^{k,p}(mathbb {R} )} for 1⩽p⩽{displaystyle 1leqslant pleqslant infty }{displaystyle 1leqslant pleqslant infty } is defined as the subset of functions f{displaystyle f}f in Lp(R){displaystyle L^{p}(mathbb {R} )}{displaystyle L^{p}(mathbb {R} )} such that f{displaystyle f}f and its weak derivatives up to order k{displaystyle k}k have a finite Lp 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)},}{displaystyle f^{(k-1)},} the (k−1){displaystyle (k-1)}(k-1)–th derivative of the function f,{displaystyle f,}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}}.}|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,dt right )^{frac{1}{p}}.

Equipped with the norm k,p,Wk,p{displaystyle |cdot |_{k,p},W^{k,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}}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}.}{displaystyle H^{k}=W^{k,2}.}

The space Hk{displaystyle H^{k}}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}}{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}}}{widehat {f}} is the Fourier series of f,{displaystyle f,}f, and T{displaystyle mathbb {T} }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}.}|f|^2_{k,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}}H^{k} admits an inner product, like the space H0=L2.{displaystyle H^{0}=L^{2}.}{displaystyle H^{0}=L^{2}.} In fact, the Hk{displaystyle H^{k}}H^{k} inner product is defined in terms of the L2{displaystyle L^{2}}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}}.}langle u,vrangle_{H^k} = sum_{i=0}^k left langle D^i u,D^i v right rangle_{L^2}.

The space Hk{displaystyle H^{k}}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)}{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)}{displaystyle W^{1,infty }(I)} is the space of Lipschitz functions on I, for every interval I. All spaces Wk,∞{displaystyle W^{k,infty }}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 .}{displaystyle p<infty .} (E.g., functions behaving like |x|−1/3 at the origin are in L2,{displaystyle L^{2},}{displaystyle L^{2},} but the product of two such functions is not in L2{displaystyle L^{2}}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)}}{displaystyle f^{(k-1)}} be the integral of f(k){displaystyle f^{(k)}}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 .}{displaystyle kin mathbb {N} ,1leqslant pleqslant infty .} The Sobolev space Wk,p(Ω){displaystyle W^{k,p}(Omega )}W^{{k,p}}(Omega ) is defined to be the set of all functions f{displaystyle f}f on Ω{displaystyle Omega }Omega such that for every multi-index α{displaystyle alpha }alpha with |⩽k,{displaystyle |alpha |leqslant k,}{displaystyle |alpha |leqslant k,} the mixed partial derivative


f(α)=∂|f∂x1α1…xnαn{displaystyle f^{(alpha )}={frac {partial ^{|alpha |}f}{partial x_{1}^{alpha _{1}}dots partial x_{n}^{alpha _{n}}}}}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 ),}{displaystyle L^{p}(Omega ),} i.e.


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

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


Wk,p(Ω)={u∈Lp(Ω):Dαu∈Lp(Ω)∀|⩽k}.{displaystyle W^{k,p}(Omega )=left{uin L^{p}(Omega ):D^{alpha }uin L^{p}(Omega ),,forall |alpha |leqslant kright}.}{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}k is called the order of the Sobolev space Wk,p(Ω).{displaystyle W^{k,p}(Omega ).}{displaystyle W^{k,p}(Omega ).}


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


u‖Wk,p(Ω):={(∑|⩽k‖Dαu‖Lp(Ω)p)1p1⩽p<∞;max|α|⩽k‖Dαu‖L∞)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}}}{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)′:={∑|⩽k‖Dαu‖Lp(Ω)1⩽p<∞;∑|⩽k‖Dαu‖L∞)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}}}{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 )}W^{{k,p}}(Omega ) is a Banach space. For p<∞,Wk,p(Ω){displaystyle p<infty ,W^{k,p}(Omega )}{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 )}{displaystyle W^{k,2}(Omega )} by Hk(Ω){displaystyle H^{k}(Omega )}H^{k}(Omega ) for it is a Hilbert space with the norm Wk,2(Ω){displaystyle |cdot |_{W^{k,2}(Omega )}}{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 )}{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}p is finite and Ω{displaystyle Omega }Omega is open, then there exists for any u∈Wk,p(Ω){displaystyle uin W^{k,p}(Omega )}{displaystyle uin W^{k,p}(Omega )} an approximating sequence of functions um∈C∞){displaystyle u_{m}in C^{infty }(Omega )}{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.} left | u_m - u right |_{W^{k,p}(Omega)} to 0.

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



Examples


In higher dimensions, it is no longer true that, for example, W1,1{displaystyle W^{1,1}}W^{{1,1}} contains only continuous functions. For example, |x|−1∈W1,1(B3){displaystyle |x|^{-1}in W^{1,1}(mathbb {B} ^{3})}{displaystyle |x|^{-1}in W^{1,1}(mathbb {B} ^{3})} where B3{displaystyle mathbb {B} ^{3}}{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 )}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 }}{displaystyle f:mathbb {B} ^{n}to mathbb {R} cup {infty }} defined on the n-dimensional ball we have:


f(x)=|x|−αWk,p(Bn)⟺α<np−k.{displaystyle f(x)=|x|^{-alpha }in W^{k,p}(mathbb {B} ^{n})Longleftrightarrow alpha <{tfrac {n}{p}}-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 .}{displaystyle 1leqslant pleqslant infty .} If a function is in W1,p(Ω),{displaystyle W^{1,p}(Omega ),}{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}}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 ).}{displaystyle L^{p}(Omega ).} Conversely, if the restriction of f{displaystyle f}f to almost every line parallel to the coordinate directions is absolutely continuous, then the pointwise gradient f{displaystyle nabla f}nabla f exists almost everywhere, and f{displaystyle f}f is in W1,p(Ω){displaystyle W^{1,p}(Omega )}{displaystyle W^{1,p}(Omega )} provided f,|∇f|∈Lp(Ω).{displaystyle f,|nabla f|in L^{p}(Omega ).}{displaystyle f,|nabla f|in L^{p}(Omega ).} In particular, in this case the weak partial derivatives of f{displaystyle f}f and pointwise partial derivatives of f{displaystyle f}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.}{displaystyle p>n.} A function in W1,p(Ω){displaystyle W^{1,p}(Omega )}{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}},}{displaystyle gamma =1-{tfrac {n}{p}},} by Morrey's inequality. In particular, if p=∞,{displaystyle p=infty ,}{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 )}{displaystyle W^{1,2}(Omega )} is also denoted by H1(Ω).{displaystyle H^{1}(Omega ).}{displaystyle H^{1}(Omega ).} It is a Hilbert space, with an important subspace H01(Ω){displaystyle H_{0}^{1}(Omega )}H_{0}^{1}(Omega ) defined to be the closure of the infinitely differentiable functions compactly supported in Ω{displaystyle Omega }Omega in H1(Ω).{displaystyle H^{1}(Omega ).}{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}}.}|f|_{H^1} = left ( int_Omega left ( |f|^2 + |nabla f|^2 right) right)^{frac{1}{2}}.

When Ω{displaystyle Omega }Omega has a regular boundary, H01(Ω){displaystyle H_{0}^{1}(Omega )}H_{0}^{1}(Omega ) can be described as the space of functions in H1(Ω){displaystyle H^{1}(Omega )}H^{1}(Omega ) that vanish at the boundary, in the sense of traces (see below). When n=1,{displaystyle n=1,}n=1, if Ω=(a,b){displaystyle Omega =(a,b)}{displaystyle Omega =(a,b)} is a bounded interval, then H01(a,b){displaystyle H_{0}^{1}(a,b)}{displaystyle H_{0}^{1}(a,b)} consists of continuous functions on [a,b]{displaystyle [a,b]}[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]}f(x) = int_a^x f'(t) , mathrm{d}t, qquad x in [a, b]

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


When Ω{displaystyle Omega }Omega is bounded, the Poincaré inequality states that there is a constant C=C(Ω){displaystyle C=C(Omega )}{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 ).}{displaystyle int _{Omega }|f|^{2}leqslant C^{2}int _{Omega }|nabla f|^{2},qquad fin H_{0}^{1}(Omega ).}

When Ω{displaystyle Omega }Omega is bounded, the injection from H01(Ω){displaystyle H_{0}^{1}(Omega )}H_{0}^{1}(Omega ) to L2(Ω),{displaystyle L^{2}(Omega ),}{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 )}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 }}u|_{partialOmega}. However, it is not clear how to describe values at the boundary for u ∈ Wk,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 )}T: W^{1,p}(Omega)to L^p(partialOmega) 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}}}{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 )}{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 ).}{displaystyle W^{1-{frac {1}{p}},p}(partial Omega ).}


Intuitively, taking the trace costs 1/p of a derivative. The functions u in W1,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},} W_0^{1,p}(Omega)= left {uin W^{1,p}(Omega): Tu=0 right },

where


W01,p(Ω):={u∈W1,p(Ω):∃{um}m=1∞Cc∞), 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}.}{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 )}{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})}{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},}{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})}.}{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}}{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 )}{displaystyle H^{s,p}(Omega )} is the set of restrictions of functions from Hs,p(Rn){displaystyle H^{s,p}(mathbb {R} ^{n})}{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}}{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, Hs,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 Wk,p(Ω) = Hk,p(Ω) holds in the sense of equivalent norms, if Ω is domain with uniform Ck-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}{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})}{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}).}{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}),}{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 .}{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 Lp-setting.[6] For 1⩽p<∞(0,1){displaystyle 1leqslant p<infty ,theta in (0,1)}{displaystyle 1leqslant p<infty ,theta in (0,1)} and f∈Lp(Ω),{displaystyle fin L^{p}(Omega ),}{displaystyle fin L^{p}(Omega ),} the Slobodeckij seminorm (roughly analogous to the Hölder seminorm) is defined by


[f]θ,p,Ω:=(∫ΩΩ|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}}.}{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)}theta = s - lfloor s rfloor 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 )}{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}.}{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 }.}{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 }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.}{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 )}{displaystyle W^{1,p}(Omega )} is not even a vector subspace of Ws,p(Ω){displaystyle W^{s,p}(Omega )}{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 )}{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 }{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 }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 }Omega to functions of Rn{displaystyle mathbb {R} ^{n}}mathbb {R} ^{n} such that:




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


  2. A:Wk,p(Ω)→Wk,p(Rn){displaystyle A:W^{k,p}(Omega )to W^{k,p}(mathbb {R} ^{n})}{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 .}Omega .



Case of p = 2


Extension operators are the most natural way to define Hs(Ω){displaystyle H^{s}(Omega )}{displaystyle H^{s}(Omega )} for non-integer s (we cannot work directly on Ω{displaystyle Omega }Omega since taking Fourier transform is a global operation). We define Hs(Ω){displaystyle H^{s}(Omega )}{displaystyle H^{s}(Omega )} by saying that u∈Hs(Ω){displaystyle uin H^{s}(Omega )}{displaystyle uin H^{s}(Omega )} if and only if Au∈Hs(Rn).{displaystyle Auin H^{s}(mathbb {R} ^{n}).}{displaystyle Auin H^{s}(mathbb {R} ^{n}).} Equivalently, complex interpolation yields the same Hs(Ω){displaystyle H^{s}(Omega )}{displaystyle H^{s}(Omega )} spaces so long as Ω{displaystyle Omega }Omega has an extension operator. If Ω{displaystyle Omega }Omega does not have an extension operator, complex interpolation is the only way to obtain the Hs(Ω){displaystyle H^{s}(Omega )}{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 )}{displaystyle H_{0}^{s}(Omega )} to be the closure in Hs(Ω){displaystyle H^{s}(Omega )}{displaystyle H^{s}(Omega )} of the space Cc∞){displaystyle C_{c}^{infty }(Omega )}{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 }Omega be uniformly Cm regular, ms and let P be the linear map sending u in Hs(Ω){displaystyle H^{s}(Omega )}{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}}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}}H^s_0 is precisely the kernel of P.


If u∈H0s(Ω){displaystyle uin H_{0}^{s}(Omega )}{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})}{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}}}{displaystyle {tilde {u}}(x)={begin{cases}u(x)&xin Omega \0&{text{else}}end{cases}}}


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

For f ∈ Lp(Ω) its extension by zero,


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

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


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

In the case of the Sobolev space W1,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}).}{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}),}{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}{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 )}.}{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}.}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}}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 }}W^{k,infty} is defined to be the Hölder space Cn where k = n + α and 0 < α ≤ 1.) The Sobolev embedding theorem states that if k⩾m{displaystyle kgeqslant m}{displaystyle kgeqslant m} and k−np⩾m−nq{displaystyle k-{tfrac {n}{p}}geqslant m-{tfrac {n}{q}}}{displaystyle k-{tfrac {n}{p}}geqslant m-{tfrac {n}{q}}} then


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

and the embedding is continuous. Moreover, if k>m{displaystyle k>m}{displaystyle k>m} and k−np>m−nq{displaystyle k-{tfrac {n}{p}}>m-{tfrac {n}{q}}}{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 }}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 Lp 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}}mathbb {R} ^{n} (Stein 1970). Sobolev embeddings on Rn{displaystyle mathbb {R} ^{n}}mathbb {R} ^{n} that are not compact often have a related, but weaker, property of cocompactness.



Notes





  1. ^ Evans 1998, Chapter 5.2


  2. ^ abc Adams 1975


  3. ^ Bergh & Löfström 1976


  4. ^ ab Triebel 1995


  5. ^ Bessel potential spaces with variable integrability have been independently introduced by Almeida & Samko (A. Almeida and S. Samko, "Characterization of Riesz and Bessel potentials on variable Lebesgue spaces", 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).


  6. ^ Lunardi 1995


  7. ^ 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: 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).




References




  • Adams, Robert A.; Fournier, John (2003) [1975]. Sobolev Spaces. Pure and Applied Mathematics. 140 (2nd ed.). Boston, MA: Academic Press. ISBN 978-0-12-044143-3..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.


  • Aubin, Thierry (1982), Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 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, 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. 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. 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, 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.


  • Nikol'skii, S.M. (2001) [1994], "Imbedding theorems", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.


  • Nikol'skii, S.M. (2001) [1994], "Sobolev space", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.


  • Sobolev, S.L. (1963), "On a theorem of functional analysis", Transl. Amer. Math. Soc., 34 (2): 39–68; 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, 120, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1015-3, ISBN 978-0-387-97017-2, MR 1014685.



External links


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



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