Sobolev inequality
Sobolev inequality (Redirected from Sobolev embedding theorem) Jump to navigation Jump to search In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev.
Contents 1 Sobolev embedding theorem 1.1 Generalizations 1.2 Kondrachov embedding theorem 2 Gagliardo–Nirenberg–Sobolev inequality 3 Hardy–Littlewood–Sobolev lemma 4 Morrey's inequality 5 General Sobolev inequalities 5.1 k < n/p 5.2 k > n/p 6 Case p=n, k=1 7 Nash inequality 8 Logarithmic Sobolev inequality 9 References Sobolev embedding theorem Graphical representation of the embedding conditions. The space W 3,p, represented by a blue dot at the point (1/p, 3), embeds into the spaces indicated by red dots, all lying on a line with slope n. The white circle at (0,0) indicates the impossibility of optimal embeddings into L ∞.
Let W k,p(Rn) denote the Sobolev space consisting of all real-valued functions on Rn whose first k weak derivatives are functions in Lp. Here k is a non-negative integer and 1 ≤ p < ∞. The first part of the Sobolev embedding theorem states that if k > ℓ, p < n and 1 ≤ p < q < ∞ are two real numbers such that {displaystyle {frac {1}{p}}-{frac {k}{n}}={frac {1}{q}}-{frac {ell }{n}},} then {displaystyle W^{k,p}(mathbf {R} ^{n})subseteq W^{ell ,q}(mathbf {R} ^{n})} and the embedding is continuous. In the special case of k = 1 and ℓ = 0, Sobolev embedding gives {displaystyle W^{1,p}(mathbf {R} ^{n})subseteq L^{p^{*}}(mathbf {R} ^{n})} where p∗ is the Sobolev conjugate of p, given by {displaystyle {frac {1}{p^{*}}}={frac {1}{p}}-{frac {1}{n}}.} This special case of the Sobolev embedding is a direct consequence of the Gagliardo–Nirenberg–Sobolev inequality. The result should be interpreted as saying that if a function {displaystyle f} in {displaystyle L^{p}(mathbb {R} ^{n})} has one derivative in {displaystyle L^{p}} , then {displaystyle f} itself has improved local behavior, meaning that it belongs to the space {displaystyle L^{p^{*}}} where {displaystyle p^{*}>p} . (Note that {displaystyle 1/p^{*}<1/p} , so that {displaystyle p^{*}>p} .) Thus, any local singularities in {displaystyle f} must be more mild than for a typical function in {displaystyle L^{p}} .
If the line from the picture above intersects the y-axis at s = r + α, the embedding into a Hölder space C r, α (red) holds. White circles indicate intersection points at which optimal embeddings are not valid.
The second part of the Sobolev embedding theorem applies to embeddings in Hölder spaces C r,α(Rn). If n < pk and {displaystyle {frac {1}{p}}-{frac {k}{n}}=-{frac {r+alpha }{n}},{mbox{ or, equivalently, }}r+alpha =k-{frac {n}{p}}} with α ∈ (0, 1) then one has the embedding {displaystyle W^{k,p}(mathbf {R} ^{n})subset C^{r,alpha }(mathbf {R} ^{n}).} This part of the Sobolev embedding is a direct consequence of Morrey's inequality. Intuitively, this inclusion expresses the fact that the existence of sufficiently many weak derivatives implies some continuity of the classical derivatives. If {displaystyle alpha =1} then {displaystyle W^{k,p}(mathbf {R} ^{n})subset C^{r,gamma }(mathbf {R} ^{n})} for every {displaystyle gamma in (0,1)} . In particular, as long as {displaystyle pk>n} , the embedding criterion will hold with {displaystyle r=0} and some positive value of {displaystyle alpha } . That is, for a function {displaystyle f} on {displaystyle mathbb {R} ^{n}} , if {displaystyle f} has {displaystyle k} derivatives in {displaystyle L^{p}} and {displaystyle pk>n} , then {displaystyle f} will be continuous (and actually Hölder continuous with some positive exponent {displaystyle alpha } ).
Generalizations The Sobolev embedding theorem holds for Sobolev spaces W k,p(M) on other suitable domains M. In particular (Aubin 1982, Chapter 2; Aubin 1976), both parts of the Sobolev embedding hold when M is a bounded open set in Rn with Lipschitz boundary (or whose boundary satisfies the cone condition; Adams 1975, Theorem 5.4) M is a compact Riemannian manifold M is a compact Riemannian manifold with boundary and the boundary is Lipschitz (meaning that the boundary can be locally represented as a graph of a Lipschitz continuous function). M is a complete Riemannian manifold with injectivity radius δ > 0 and bounded sectional curvature.
If M is a bounded open set in Rn with continuous boundary, then W 1,2(M) is compactly embedded in L2(M) (Nečas 2012, Section 1.1.5, Theorem 1.4).
Kondrachov embedding theorem Main article: Rellich–Kondrachov theorem On a compact manifold M with C1 boundary, the Kondrachov embedding theorem states that if k > ℓ and {displaystyle {frac {1}{p}}-{frac {k}{n}}<{frac {1}{q}}-{frac {ell }{n}}} then the Sobolev embedding {displaystyle W^{k,p}(M)subset W^{ell ,q}(M)} is completely continuous (compact).[1] Note that the condition is just as in the first part of the Sobolev embedding theorem, with the equality replaced by an inequality, thus requiring a more regular space W k,p(M). Gagliardo–Nirenberg–Sobolev inequality Assume that u is a continuously differentiable real-valued function on Rn with compact support. Then for 1 ≤ p < n there is a constant C depending only on n and p such that {displaystyle |u|_{L^{p^{*}}(mathbf {R} ^{n})}leq C|Du|_{L^{p}(mathbf {R} ^{n})}.} with 1/p* = 1/p - 1/n. The case {displaystyle 1
The Hardy–Littlewood–Sobolev lemma implies the Sobolev embedding essentially by the relationship between the Riesz transforms and the Riesz potentials.
Morrey's inequality Assume n < p ≤ ∞. Then there exists a constant C, depending only on p and n, such that {displaystyle |u|_{C^{0,gamma }(mathbf {R} ^{n})}leq C|u|_{W^{1,p}(mathbf {R} ^{n})}} for all u ∈ C1(Rn) ∩ Lp(Rn), where {displaystyle gamma =1-{frac {n}{p}}.} Thus if u ∈ W 1,p(Rn), then u is in fact Hölder continuous of exponent γ, after possibly being redefined on a set of measure 0. A similar result holds in a bounded domain U with Lipschitz boundary. In this case, {displaystyle |u|_{C^{0,gamma }(U)}leq C|u|_{W^{1,p}(U)}} where the constant C depends now on n, p and U. This version of the inequality follows from the previous one by applying the norm-preserving extension of W 1,p(U) to W 1,p(Rn). The inequality is named after Charles B. Morrey Jr. General Sobolev inequalities Let U be a bounded open subset of Rn, with a C1 boundary. (U may also be unbounded, but in this case its boundary, if it exists, must be sufficiently well-behaved.) Assume u ∈ W k,p(U). Then we consider two cases: k < n/p In this case we conclude that u ∈ Lq(U), where {displaystyle {frac {1}{q}}={frac {1}{p}}-{frac {k}{n}}.} We have in addition the estimate {displaystyle |u|_{L^{q}(U)}leq C|u|_{W^{k,p}(U)}} , the constant C depending only on k, p, n, and U. k > n/p Here, we conclude that u belongs to a Hölder space, more precisely: {displaystyle uin C^{k-left[{frac {n}{p}}right]-1,gamma }(U),} where {displaystyle gamma ={begin{cases}left[{frac {n}{p}}right]+1-{frac {n}{p}}&{frac {n}{p}}notin mathbf {Z} \{text{any element in }}(0,1)&{frac {n}{p}}in mathbf {Z} end{cases}}} We have in addition the estimate {displaystyle |u|_{C^{k-left[{frac {n}{p}}right]-1,gamma }(U)}leq C|u|_{W^{k,p}(U)},} the constant C depending only on k, p, n, γ, and U. In particular, the condition {displaystyle k>n/p} guarantees that {displaystyle u} is continuous (and actually Hölder continuous with some positive exponent).
Case p=n, k=1 If {displaystyle uin W^{1,n}(mathbf {R} ^{n})} , then u is a function of bounded mean oscillation and {displaystyle |u|_{BMO}leq C|Du|_{L^{n}(mathbf {R} ^{n})},} for some constant C depending only on n. This estimate is a corollary of the Poincaré inequality.
Nash inequality The Nash inequality, introduced by John Nash (1958), states that there exists a constant C > 0, such that for all u ∈ L1(Rn) ∩ W 1,2(Rn), {displaystyle |u|_{L^{2}(mathbf {R} ^{n})}^{1+2/n}leq C|u|_{L^{1}(mathbf {R} ^{n})}^{2/n}|Du|_{L^{2}(mathbf {R} ^{n})}.} The inequality follows from basic properties of the Fourier transform. Indeed, integrating over the complement of the ball of radius ρ, {displaystyle int _{|x|geq rho }left|{hat {u}}(x)right|^{2},dxleq int _{|x|geq rho }{frac {|x|^{2}}{rho ^{2}}}left|{hat {u}}(x)right|^{2},dxleq rho ^{-2}int _{mathbf {R} ^{n}}|Du|^{2},dx} (1) because {displaystyle 1leq |x|^{2}/rho ^{2}} . On the other hand, one has {displaystyle |{hat {u}}|leq |u|_{L^{1}}} which, when integrated over the ball of radius ρ gives {displaystyle int _{|x|leq rho }|{hat {u}}(x)|^{2},dxleq rho ^{n}omega _{n}|u|_{L^{1}}^{2}} (2) where ωn is the volume of the n-ball. Choosing ρ to minimize the sum of (1) and (2) and applying Parseval's theorem: {displaystyle |{hat {u}}|_{L^{2}}=|u|_{L^{2}}} gives the inequality.
In the special case of n = 1, the Nash inequality can be extended to the Lp case, in which case it is a generalization of the Gagliardo-Nirenberg-Sobolev inequality (Brezis 2011, Comments on Chapter 8). In fact, if I is a bounded interval, then for all 1 ≤ r < ∞ and all 1 ≤ q ≤ p < ∞ the following inequality holds {displaystyle |u|_{L^{p}(I)}leq C|u|_{L^{q}(I)}^{1-a}|u|_{W^{1,r}(I)}^{a},} where: {displaystyle aleft({frac {1}{q}}-{frac {1}{r}}+1right)={frac {1}{q}}-{frac {1}{p}}.} Logarithmic Sobolev inequality Main article: Logarithmic Sobolev inequalities The simplest of the Sobolev embedding theorems, described above, states that if a function {displaystyle f} in {displaystyle L^{p}(mathbb {R} ^{n})} has one derivative in {displaystyle L^{p}} , then {displaystyle f} itself is in {displaystyle L^{p^{*}}} , where {displaystyle 1/p^{*}=1/p-1/n.} We can see that as {displaystyle n} tends to infinity, {displaystyle p^{*}} approaches {displaystyle p} . Thus, if the dimension {displaystyle n} of the space on which {displaystyle f} is defined is large, the improvement in the local behavior of {displaystyle f} from having a derivative in {displaystyle L^{p}} is small ( {displaystyle p^{*}} is only slightly larger than {displaystyle p} ). In particular, for functions on an infinite-dimensional space, we cannot expect any direct analog of the classical Sobolev embedding theorems. There is, however, a type of Sobolev inequality, established by Leonard Gross (Gross 1975) and known as a logarithmic Sobolev inequality, that has dimension-independent constants and therefore continues to hold in the infinite-dimensional setting. The logarithmic Sobolev inequality says, roughly, that if a function is in {displaystyle L^{p}} with respect to a Gaussian measure and has one derivative that is also in {displaystyle L^{p}} , then {displaystyle f} is in " {displaystyle L^{p}} -log", meaning that the integral of {displaystyle |f|^{p}log |f|} is finite. The inequality expressing this fact has constants that do not involve the dimension of the space and, thus, the inequality holds in the setting of a Gaussian measure on an infinite-dimensional space. It is now known that logarithmic Sobolev inequalities hold for many different types of measures, not just Gaussian measures. Although it might seem as if the {displaystyle L^{p}} -log condition is a very small improvement over being in {displaystyle L^{p}} , this improvement is sufficient to derive an important result, namely hypercontractivity for the associated Dirichlet form operator. This result means that if a function is in the range of the exponential of the Dirichlet form operator—which means that the function has, in some sense, infinitely many derivatives in {displaystyle L^{p}} —then the function does belong to {displaystyle L^{p^{*}}} for some {displaystyle p^{*}>p} (Gross 1975 Theorem 6).
References ^ Taylor, Michael E. (1997). Partial Differential Equations I - Basic Theory (2nd ed.). p. 286. ISBN 0-387-94653-5. Adams, Robert A. (1975), Sobolev Spaces, Pure and Applied Mathematics, vol. 65, Academic Press, ISBN 978-0-12-044150-1, MR 0450957. Aubin, Thierry (1976), "Espaces de Sobolev sur les variétés riemanniennes", Bulletin des Sciences Mathématiques, 2e Série, 100 (2): 149–173, MR 0488125 Aubin, Thierry (1982), Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, doi:10.1007/978-1-4612-5734-9, ISBN 978-0-387-90704-8, MR 0681859. Brezis, Haïm (1983), Analyse Fonctionnelle: théorie et applications, Paris: Masson, ISBN 0-8218-0772-2 Brezis, Haïm (2011), Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer Science & Business Media, ISBN 978-0-387-70913-0 Evans, Lawrence (1998), Partial Differential Equations, Providence RI: American Mathematical Society, ISBN 0-8218-0772-2 Gross, Leonard (1975), "Logarithmic Sobolev inequalities", American Journal of Mathematics, 97 (4): 1061–1083, doi:10.2307/2373688, JSTOR 2373688 Leoni, Giovanni (2009), A First Course in Sobolev Spaces, Graduate Studies in Mathematics, American Mathematical Society, ISBN 978-0-8218-4768-8 MR2527916, Zbl 1180.46001, MAA review Maz'ja, Vladimir G. (1985), Sobolev Spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Translated from the Russian by T. O. Shaposhnikova. Nash, J. (1958), "Continuity of solutions of parabolic and elliptic equations", American Journal of Mathematics, 80 (4): 931–954, doi:10.2307/2372841, hdl:10338.dmlcz/101876, JSTOR 2372841. Nečas, J. (2012), Direct Methods in the Theory of Elliptic Equations, Springer Monographs in Mathematics. Nikol'skii, S.M. (2001) [1994], "Imbedding theorems", Encyclopedia of Mathematics, EMS Press Schikorra, Armin; Spector, Daniel; Van Schaftingen, Jean (2017), "An {displaystyle L^{1}} -type estimate for Riesz potentials", Revista Matemática Iberoamericana, 33 (1): 291–304, arXiv:1411.2318, doi:10.4171/rmi/937, S2CID 55497245 Stein, Elias (1970), Singular Integrals and Differentiability Properties of Functions, Princeton, NJ: Princeton University Press, ISBN 0-691-08079-8 hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: InequalitiesSobolev spacesCompactness theorems
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