Basic properties of Sobolev's spaces on time scales

BASIC PROPERTIES OF SOBOLEV’S SPACES ON TIME SCALES RAVI P. AGARWAL, VICTORIA OTERO–ESPINAR, KANISHKA PERERA, AND DOLORES R. VIVERO Received 18 January 2006; Accepted 22 January 2006 We study the theory of Sobolev’s spaces of functions defined on a closed subinterval of an arbitrary time scale endowed with the Lebesgue Δ-measure; analogous properties to that valid for Sobolev’s spaces of functions defined on an arbitrary open interval of the real numbers are derived. Copyright © 2006 Ravi P. Agarwal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Sobolev’s spaces are a fundamental tool in real analysis, for instance, in the use of variational methods to solve boundary value problems in ordinary and partial differential equations and difference equations. In spite of this, theory for functions defined on an arbitrary bounded open interval of the real numbers is well known, see [2], and for functions defined on an arbitrary bounded subset of the natural numbers is trivial, as far as we know, for functions defined on an arbitrary time scale, it has not been studied before. The aim of this paper is to give an introduction to Sobolev’s spaces of functions defined on a closed interval [a,b]∩Tof an arbitrary time scale Tendowed with the Lebesgue Δ- measure. In Section 2, we gather together the concepts one needs to read this paper, such as the Lpspaces linked to the Lebesgue Δ-measure and absolutely continuous functions on an arbitrary closed interval of T. The most important part of this paper is Section 3 where we define the first-order Sobolev’s spaces as the space of Lp Δ([a,b)∩T) functions whose generalized Δ-derivative belongs to Lp Δ([a,b)∩T), moreover, we study some of their properties by establishing an equivalence between them and the usual Sobolev’s spaces defined on an open interval of the real numbers. Section 4 is devoted to the generalization of Sobolev’s spaces to order n≥2. 2. Preliminaries The Lebesgue Δ-measure μΔwas defined in [1, Section 5.7] or in [5, Section 5] as the Carath´ eodory extension of a set function and it may be characterized in terms of Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 38121, Pages 1–14 DOI 10.1155/ADE/2006/38121 2 Basic properties of Sobolev’s spaces on time scales well-known measures as the following result shows; we refer the reader to [6–8]fora broad introduction to measure and integration theory. Proposition 2.1. The Lebesgue Δ-measure is defined over the Lebesgue measurable subsets of T; moreover, it satisfies the following equality: μΔ=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ λ+ i∈Iσti−ti·δti+μM,if M∈T, λ+ i∈Iσti−ti·δti,if M∈ T,(2.1) where {ti}i∈I,I⊂N, is the set of all right-scattered points of T,Mis the supremum of T,λ is the Lebesgue measure, δtiis the Dirac measure concentrate at ti,andμMis a degenerate measure defined as μM(A)=0if M∈ Aand μM(A)=+∞if M∈A. Proof. From properties of measure, one can deduce relation (2.1) for the outer measures linked to these measures which plainly yields to (2.1).  As a straightforward consequence of equality (2.1), one can deduce the following formula to calculate the Lebesgue Δ-integral; this formula was proved in [4], nevertheless, we remark that this argument is more simple than that. Proposition 2.2. Let E⊂Tbe a Δ-measurable set. If f:T→Ris Δ-integrableonE, then Ef(s)Δs=Ef(s)ds+ i∈IEσti−ti·fti+r(f,E), (2.2) where r(f,E)=⎧ ⎨ ⎩ μM(E)·f(M), if M∈T, 0, if M∈ T,(2.3) IE:={i∈I:ti∈E}and {ti}i∈I,I⊂N, is the set of all right-scattered points of T. Definition 2.3. Let A⊂T.Ais called Δ-null set if μΔ(A)=0. Say that a property Pholds Δ-almost everywhere (Δ-a.e.) on A,orforΔ-almost all (Δ-a.a.) t∈Aif there is a Δ-null set E⊂Asuch that Pholds for all t∈A\E. Definition 2.4. Let E⊂Tbe a Δ-measurable set and let p∈¯ R≡[−∞,+∞]besuchthat p≥1andlet f:E→¯ Rbe a Δ-measurable function. Say that fbelongs to Lp Δ(E)provided that either E |f|p(s)Δs<∞if p∈R, (2.4) or there exists a constant C∈Rsuch that |f|≤CΔ-a.e. on Eif p=+∞.(2.5) Note that equality (2.2) guarantees that in order for f:T→Rto belong to Lp Δ(T), p∈R,andTbounded from above, it is necessary that f(M)=0. We will work with the Ravi P. Agarwal et al. 3 Lp Δ(Jo)spaces,whereJ=[a,b]∩T,a,b∈T,a<b, is an arbitrary closed subinterval of T and Jo=[a,b)∩T; we state some of their properties whose proofs can be found in [6–8]. Theorem 2.5. Let p∈¯ Rbe such that p≥1.Then,thesetLp Δ(Jo)is a Banach space together with the norm defined for every f∈Lp Δ(Jo)as fLp Δ:=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Jo |f|p(s)Δs1/p ,if p∈R, inf C∈R:|f|≤CΔ-a.e. on Jo,if p=+∞. (2.6) Moreover, L2 Δ(Jo)is a Hilbert space together with the inner product given for every (f,g)∈ L2 Δ(Jo)×L2 Δ(Jo)by (f,g)L2 Δ:=Jof(s)·g(s)Δs. (2.7) Proposition 2.6. Suppose p∈¯ Rand p≥1.Letp∈¯ Rbe such that 1/p+1/p=1. Then, if f∈Lp Δ(Jo)and g∈Lp Δ(Jo), then f·g∈L1 Δ(Jo)and f·gL1 Δ≤fLp Δ·gLp Δ.(2.8) This expression is called H¨ older’s inequality and Cauchy-Schwarz’s inequality whenever p=2. Proposition 2.7. If p∈Rand p≥1,then,thesetCc(Jo)of all continuous functions on Jo with compact support in Jois dense in Lp Δ(Jo). As a consequence of Proposition 2.2, one can establish the following equivalence between the Lp Δ(Jo) spaces and the usual Lp([a,b]) spaces linked to the Lebesgue measure. Corollary 2.8. Let p∈¯ Rwith p≥1,let f:J→¯ R,andlet  f:[a,b]→¯ Rbe the extension of fto [a,b]defined as  f(t):=⎧ ⎨ ⎩ f(t), if t∈J, f(ti), if t∈ti,σti,for some i∈IJ,(2.9) with IJ:={i∈I:ti∈J}and {ti}i∈I,I⊂N, is the set of all right-scattered points of T. Then, f∈Lp Δ(Jo)if and only if  f∈Lp([a,b]). In this case, fLp Δ= fLp.(2.10) As we know from general theory of Sobolev’s spaces, another important class of functions is just the absolutely continuous functions. Definition 2.9. A function f:J→Ris said to be absolutely continuous on J,f∈AC(J), if for every ε>0, there exists a δ>0suchthatif{[ak,bk)∩T}n k=1,withak,bk∈J,is a finite pairwise disjoint family of subintervals of Jsatisfying n k=1(bk−ak)<δ,then n k=1|f(bk)−f(ak)|<ε. 4 Basic properties of Sobolev’s spaces on time scales These functions are precisely that for which the fundamental theorem of Calculus holds. Theorem 2.10 [3, Theorem 4.1]. Afunction f:J→Ris absolutely continuous on Jif and only if fis Δ-differentiable Δ-a.e. on Jo,fΔ∈L1 Δ(Jo)and f(t)=f(a)+[a,t)∩T fΔ(s)Δs,∀t∈J. (2.11) Absolutely continuous functions on Tverify the integration by parts formula. Theorem 2.11. If f,g:J→Rare absolutely continuous functions on J, then f·gis absolutely continuous on Jand the following equality is valid: JofΔg+fσgΔ(s)Δs=f(b)g(b)−f(a)g(a)=JofgΔ+fΔgσ(s)Δs. (2.12) They are linked to the class of absolutely continuous functions on [a,b]asthefollowing property shows. Corollary 2.12 [3, Corollary 3.1]. Assume that f:J→Rand define ¯ f:[a,b]→Ras ¯ f(t): =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ f(t), if t∈J, fti+fσti−fti σti−tit−ti,if t∈ti,σti,for some i∈IJ,(2.13) with IJ:={i∈I:ti∈J}and {ti}i∈I,I⊂N, is the set of all right-scattered points of T. Then, fis absolutely continuous on Jif and only if ¯ fis absolutely continuous on [a,b]. Moreover, for every n∈N,n≥1, we will denote as ACn(J):=x∈AC(J):xΔj∈ACJκj∀j∈{1,...,n}, (2.14) where for every j∈N,j≥1, Jκj=[a,ρj(b)] ∩T. 3. First-order Sobolev’s spaces The aim of this section is to study the first-order Sobolev’s spaces on Jequipped with the Lebesgue Δ-measure. Definition 3.1. Let p∈¯ Rbe such that p≥1andu:J→¯ R.Saythatubelongs to W1,p Δ(J) if and only if u∈Lp Δ(Jo) and there exists g:Jκ→¯ Rsuch that g∈Lp Δ(Jo)and Jou·ϕΔ(s)Δs=−Jog·ϕσ(s)Δs∀ϕ∈C1 0,rdJκ(3.1) with C1 0,rdJκ:=f:J−→ R:f∈C1 rdJκ,f(a)=0=f(b)(3.2) and C1 rd(Jκ) is the set of all continuous functions on Jsuch that they are Δ-differentiable on Jκand their Δ-derivatives are rd-continuous on Jκ. Ravi P. Agarwal et al. 5 The integration by parts formula for absolutely continuous functions on Jestablishes that the relation V1,p Δ(J):=x∈AC(J):xΔ∈Lp ΔJo⊂W1,p Δ(J) (3.3) is true for every p∈¯ Rwith p≥1. We will show that both sets are, as class of functions, equivalent; for this purpose, we need the following lemmas. Lemma 3.2. Let f∈L1 Δ(Jo)be such that the following equality is true: Jo(f·u)(s)Δs=0, ∀u∈CcJo, (3.4) then f≡0Δ-a.e. on Jo.(3.5) Proof. Fix ε>0, the density of Cc(Jo)inL1 Δ(Jo) guarantees the existence of f1∈Cc(Jo) such that f−f1L1 Δ<ε,andso,by(3.4), we deduce that for every u∈Cc(Jo), it is true that    Jof1·u(s)Δs   ≤ uC(Jo)· f−f1 L1 Δ<εuC(Jo).(3.6) Because the sets A1:=s∈Jo:f1(s)≥ε,A2:=s∈Jo:f1(s)≤−ε(3.7) are compact and disjoint subsets of Jo, Urysohn’s lemma allows to construct a function u0:Jo→Rwhich belongs to Cc(Jo)anditverifies u0≡⎧ ⎨ ⎩ 1; on A1, −1; on A2, u0 ≤1onJo; (3.8) so that, by defining A:=A1∪A2,wehavethat Jo f1 (s)Δs=Jof1·u0(s)Δs−Jo\Af1·u0(s)Δs +Jo\A f1 (s)Δs≤ε+2ε(b−a). (3.9) As a consequence of the arbitrary choice of ε>0, we achieve (3.5).  Lemma 3.3. Let f∈L1 Δ(Jo). Then, a necessary and sufficient condition for the validity of the equality Jof·ϕΔ(s)Δs=0, for every ϕ∈C1 0,rdJκ, (3.10) is the existence of a constant c∈Rsuch that f≡cΔ-a.e. on Jo.(3.11) 6 Basic properties of Sobolev’s spaces on time scales Proof. The necessary condition is consequence of the fundamental theorem of Calculus. Conversely, fix u∈Cc(Jo) arbitrary; by defining h,ϕ:J→Ras h(t): =⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ u(t)−Jou(r)Δr b−a,ift∈Jo, −Jou(r)Δr b−a,ift=b, ϕ(t): =[a,t)∩T h(s)Δs,∀t∈J, (3.12) the fundamental theorem of Calculus establishes that ϕ∈C1 0,rd(Jκ) and so, equality (3.10) yields to 0=Jof·u−Jou(r)Δr b−a(s)Δs =Jof−Jof(r)Δr b−a·u(s)Δs. (3.13) Therefore, Lemma 3.2 allows to deduce (3.11)withc=Jof(r)Δr/(b−a).  Now, we are able to prove the characterization of functions in W1,p Δ(J)intermsof functions in V1,p Δ(J). Theorem 3.4. Suppose that u∈W1,p Δ(J)for some p∈¯ Rwith p≥1and that (3.1)holds for g∈Lp Δ(Jo). Then, there exists a unique function x∈V1,p Δ(J)such that the equalities x=u,xΔ=gΔ-a.e. on Jo(3.14) are satisfied. Moreover, if g∈Crd(Jκ), then there exists a unique function x∈C1 rd(Jκ)such that x=uΔ-a.e. on Jo,xΔ=gon Jκ.(3.15) Proof. Define v:J→Ras v(t): =[a,t)∩T g(s)Δs,∀t∈J; (3.16) the fundamental theorem of Calculus guarantees that v∈V1,p Δ(J) and by the integration by parts formula, we have that for every ϕ∈C1 0,rd(Jκ), Jo(v−u)·ϕΔ(s)Δs=−JovΔ−g·ϕσ(s)Δs=0; (3.17) so that, Lemma 3.3 ensures the existence of a constant c∈Rsuch that v−u≡cΔ-almost everywhere on Jo. As a consequence of the fundamental theorem of Calculus we conclude that function x:J→Rdefined as x(t):=v(t)−cfor all t∈Jis the unique function in V1,p Δ(J) for which (3.14)isvalid. Ravi P. Agarwal et al. 7 Furthermore, if g∈Crd(Jκ), then the fundamental theorem of Calculus establishes that x∈C1 rd(Jκ)andxΔ=gon Jκ. By identifying every function in W1,p Δ(J) with its absolutely continuous representative in V1,p Δ(J) for which (3.14) holds, the set W1,p Δ(J) can be endowed with the structure of Banach space. Theorem 3.5. Assume p∈¯ Rand p≥1. The set W1,p Δ(J)is a Banach space together with the norm defined for every x∈W1,p Δ(J)as xW1,p Δ:=xLp Δ+ xΔ Lp Δ.(3.18) Moreover, the set H1 Δ(J):=W1,2 Δ(J)is a Hilbert space together with the inner product given for every (x,y)∈H1 Δ(J)×H1 Δ(J)by (x,y)H1 Δ:=(x,y)L2 Δ+xΔ,yΔL2 Δ.(3.19) Proof. Let {xn}n∈Nbe a Cauchy sequence in W1,p Δ(J); Theorem 2.5 guarantees the existence of u,g∈Lp Δ(Jo)suchthat{xn}n∈Nand {xΔ n}n∈Nconverge strongly in Lp Δ(Jo)touand g, respectively, and so, by taking limits in the equality Joxn·ϕΔ(s)Δs=−JoxΔ n·ϕσ(s)Δs,ϕ∈C1 0,rd(Jκ), (3.20) we conclude that u∈W1,p Δ(J). Thereby, it follows from Theorem 3.4, that there exists x∈W1,p Δ(J)suchthat{xn}n∈Nconverges strongly in W1,p Δ(J)tox. 3.1. Some properties. We will derive some properties of the Banach space W1,p Δ(J); the first one asserts that W1,p Δ(J) is continuously inmersed into C(J) equipped with the supremum norm · C(J). Proposition 3.6. Assume p∈¯ Rwith p≥1, then there exists a constant K>0,onlydependent on b−a, such that the inequality xC(J)≤K·xW1,p Δ(3.21) holds for all x∈W1,p Δ(J)and hence, the immersion W1,p Δ(J)C(J)is continuous. Proof. Fix x∈W1,p Δ(J). Let t,T∈Jbe such that |x(t)|:=mins∈T|x(s)|and |x(T)|:= maxs∈T|x(s)|; there is no harm in assuming t≤T. The fundamental theorem of Calculus and H¨ older’s inequality lead to xC(J)≤|x(t)|+[t,T)∩T |xΔ|(s)Δs≤K·xW1,p Δ, (3.22) for some K>0, only dependent on b−a. The strong compactness criterion in C(J)andProposition 3.6 allowtoprovethefollowing compactness property in C(J). 8 Basic properties of Sobolev’s spaces on time scales Proposition 3.7. Let p∈¯ Rbe such that p≥1. Then, the following statements are true. (1) If p>1, then the immersion W1,p Δ(J)C(J)is compact. (2) If p=1, then the immersion W1,p Δ(J)C(J)is compact if and only if every point of Jis isolated. Proof. Denote by Ᏺpthe closed unit ball in W1,p Δ(J); we know from Theorem 3.4 that Ᏺp is closed and bounded in C(J). If p>1, then the fundamental theorem of Calculus and H¨ older’s inequality ensure that Ᏺpis equicontinuous. On the other hand, if p=1, then it is clear that Ᏺpis equicontinuous whenever every point of Jis isolated, while if there exists t0∈Tsuch that t0is not isolated, then we will prove that Ᏺpis not equicontinuous. Let S:=1/(b−a+1),letδ>0bearbitraryandletsδ∈(t0−δ,t0+δ)∩Tbe such that sδ= t0; it is not a loss of generality assuming sδ<t 0. Define fδ:J→Ras fδ:=⎧ ⎪ ⎨ ⎪ ⎩ S t0−sδ ,ift∈sδ,t0∩J, 0, if t∈ sδ,t0∩J; (3.23) the fundamental theorem of Calculus asserts that Fδ:J→Rgiven by Fδ(t):=[a,t)∩T fδ(s)Δs,t∈J, (3.24) belongs to Ᏺp;sothat,as Fδt0−Fδsδ=[sδ,t0)∩T fδ(s)Δs=S, (3.25) we conclude that Ᏺpis not equicontinuous. Therefore, Arzel` a-Ascoli theorem establishes our claims.  As a consequence of Proposition 3.6, we achieve the following sufficient condition for strong convergence in C(J). Corollary 3.8. Let p∈¯ Rbe such that p>1,let{xm}m∈N⊂W1,p Δ(J),andletx∈W1,p Δ(J). If {xm}m∈Nconverges weakly in W1,p Δ(J)to x, then {xm}m∈Nconverges strongly in C(J) to x. Proof. Suppose {xm}m∈Nconverges weakly in W1,p Δ(J)tox;Proposition 3.6 establishes that {xm}m∈Nconverges weakly in C(J)toxand so, as {xm}m∈Nis equicontinuous, {xm}m∈Nconverges strongly in C(J)tox. Moreover, Proposition 3.6 allows to deduce the following equivalence between the Sobolev’s spaces on J,W1,p Δ(J), and the usual Sobolev’s spaces on (a,b), W1,p((a,b)). Corollary 3.9. Suppose that p∈¯ Rand p≥1,x:J→Rand ¯ x:[a,b]→Ris the extension of xto [a,b]defined in (2.13). Then, xbelongs to W1,p Δ(J)if and only if ¯ xbelongs to W1,p((a,b)). Ravi P. Agarwal et al. 9 Moreover, there exist two constants K1,K2>0which only depend on (b−a)such that the inequalities K1·¯ xW1,p≤xW1,p Δ ≤K2·¯ xW1,p(3.26) are satisfied for every x∈W1,p Δ(J)and p∈¯ Rwith p≥1. Proof. Let ¯ x, xΔ:[a,b]→Rbe the extensions of xand xΔto [a,b]definedin(2.13)and (2.9), respectively; it is not difficult to deduce the following equality:  xΔ=¯ xa.e. on [a,b].(3.27) Therefore, Corollaries 2.8 and 2.12 and Proposition 3.6 yield to the result.  As an application of the previous result, we will prove that some properties known for W1,p((a,b)) are directly transferred to W1,p Δ(J); in order to do this, we will use the following result. Proposition 3.10. If y:[a,b]→Rbelongs to W1,p((a,b)) for some p∈¯ Rwith p≥1, then y|Jbelongs to W1,p Δ(J). Moreover, there exists a constant T>0which only depends on (b−a)such that y|JW1,p Δ ≤T·yW1,p,∀y∈W1,p(a,b),p∈¯ R,p≥1.(3.28) Proof. Let R={ti}i∈I,I⊂N, be the set of all right-scattered points of T,letIJo={i∈I, ti∈Jo}and suppose y∈W1,p((a,b)) for some p∈¯ Rwith p≥1. The classical fundamental theorem of Calculus allows to assert that y|JΔti=[ti,σ(ti)] y(s)ds σti−ti ,foreveryi∈IJo, y|JΔ=ya.e. on Jo∩(T\R). (3.29) Therefore, if p=+∞, then it is clear that y|J∈W1,p Δ(J)and(3.28) holds while if p∈R, then, by (2.2), we have that  y|JΔ p Lp Δ≤Jo∩(T\R) y p(s)ds+ i∈IJo[ti,σ(ti)]  y p(s)ds ≤yp W1,p, (3.30) moreover,asweknowthat  y|J Lp Δ≤(b−a)1/p ·yC([a,b]) ≤C·(b−a)1/p ·yW1,p, (3.31) for some C>0, it turns out that y|J∈W1,p Δ(J)and(3.28)istrue.  Next, we deduce some properties in W1,p Δ(J) from the analogous ones in W1,p((a,b)). Corollary 3.11. Let p∈¯ Rbe such that p≥1.Then,foreveryq∈[1,+∞), the inmersion W1,p Δ(J)Lq Δ(Jo)is compact.