ON THE SOBOLEV TRACE THEOREM FOR VARIABLE EXPONENT SPACES IN THE CRITICAL RANGE. 3 JULIA´NFERNA´NDEZ BONDER,NICOLASSAINTIERANDANALIASILVA 1 0 2 Abstract. In this paper we study the Sobolev Trace Theorem for variable exponent spaces n with critical exponents. We find conditions on the best constant in order to guaranty the a existence of extremals. Then we give local conditions on the exponents and on the domain (in J thespiritofAdimurthyandYadava)inordertosatisfysuchconditions,andthereforetoensure 4 theexistence of extremals. 1 ] P A . 1. Introduction h t a The study of variable exponent Lebesgue and Sobolev spaces have deserved a great deal of m attention in the last few years due to many interesting new applications including the mathe- [ matical modeling of electrorheological fluids (see [21]) and image processing (see [3]). We refer 1 to section 2 below for a brief account of the main rsults needed here, and to the book [4] for a v complete account on these spaces. 1 6 One fundamental point in the study of these spaces is the generalization of the well–known 9 2 Sobolev immersion Theorems. That is, if Ω ⊂ RN is a bounded domain and p: Ω → [1,∞) is a . finite exponent such that sup p < N the following immersions hold 1 Ω 0 3 W1,p(x)(Ω)֒→ Lq(x)(Ω) and W1,p(x)(Ω) ֒→ Lr(x)(∂Ω), 0 1 : if the exponents q: Ω → [1,∞) and r: ∂Ω → [1,∞) verify the bounds v i Np(x) (N −1)p(x) X q(x) ≤p∗(x) := and r(x) ≤ p (x) := . ∗ r N −p(x) N −p(x) a Theseexponentsp∗(x)andp (x)arecalledthecriticalSobolevexponentandthecriticalSobolev ∗ trace exponent respectively. (Some mild regularity assumptions on the exponents are needed in order for the immersions to hold, see [4] and Section 2). These immersions can be restated as k∇vk Lp(x)(Ω) 0 < S(p(·),q(·),Ω) := inf , and v∈W1,p(x)(Ω) kvkLq(x)(Ω) 0 kvk W1,p(x)(Ω) 0 < T(p(·),r(·),Ω) := inf . v∈W1,p(x)(Ω) kvkLr(x)(Ω) Here, the norms that are considered are the Luxemburg norms. We refer to Section 2 for the precise definitions. 2000 Mathematics Subject Classification. 46E35,35B33. Keywordsandphrases. Sobolevembedding,variableexponents,criticalexponents,concentrationcompactness. 1 2 J.FERNA´NDEZBONDER,N.SAINTIERANDA.SILVA An important and interesting problem is the study of the existence of extremals for these im- mersionsi.e. functionsrealizingtheinfimuminthedefinitionofS(p(·),q(·),Ω)andT(p(·),r(·),Ω). When the exponents are uniformly subcritical, i.e. inf(p∗−q)> 0 and inf(p −r)> 0, ∗ Ω ∂Ω the immersions are compact, and so the existence of extremals follows by a direct minimization procedure. The situation when the subcriticality is violated is much more complicated. Inconstrastwiththeconstantcriticalexponentcasewhichhasdeservedalotofattentionsince Aubin’ seminal work [2], the critical immersion for variable exponent have only been considered recently. In [18], the authors study some cases where even if the subcriticality is violated, the immersion W1,p(x)(Ω) ֒→ Lq(x)(Ω) remains compact. This result requires for very restrictive 0 hypotheses on the exponents p and q, so a more general result is desirable. In this direction, in [11], applying an extension of the P.L. Lions’ Concentration–Compactness Principle for the variable exponent case (see [12, 13]) the authors proved that S(p(·),q(·),Ω) ≤ sup inf S(p(·),q(·),B (x)), ε ε>0x∈A whereA = {x ∈ Ω: q(x) = p∗(x)} is the critical set, and B (x) is the ball centered at x of radius ε ε. Moreover, in that paper it is shown that if the strict inequality holds, namely S(p(·),q(·),Ω) < sup inf S(p(·),q(·),B (x)), ε ε>0x∈A then there exists an extremal for S(p(·),q(·),Ω). Some conditions on p, q and Ω are also given in order for this strict inequality to hold. We also refer to [10] where this result is applied to obtain the existence of a solution to a critical equation involving the p(x)−Laplacian. The purpose of this article is to extend the above mentioned results to the trace problem. That is, we assume hereafter that the subcriticality for the trace exponent fails in the sense that A := {x ∈∂Ω: r(x) =p (x)} =6 ∅, T ∗ and find conditions on the exponents p, r and on the domain Ω in order to ensure the existence of an extremal for T(p(·),r(·),Ω). Up to our knowledge, this is the first paper where the critical trace inequality, in the context of variable exponent Sobolev spaces, is addressed. Concerning the constant exponent case, it is known, see [7], that k∇vk T(p,p ,Ω) ≤ K¯(N,p)−1 = inf Lp(RN+) , ∗ v∈D¯1,p(RN+)kvkLp∗(RN−1) where D¯1,p(RN) is the set of measurable functions f(y,t) such that ∂ f ∈ Lp(RN), i = 1,...,N, + i + and f(·,0) ∈ Lp∗(RN−1). Moreover, in [7] it is shown that if (1.1) T(p,p ,Ω)< K¯(N,p)−1, ∗ then there exists an extremal for the trace inequality. Notice that one trivial global condition on Ω that implies (1.1) is 1 |Ω|p (1.2) < K¯(N,p)−1, 1 HN−1(∂Ω)p∗ where Hd denotes the d−dimensional Hausdorff measure. Observe that the family of sets veri- fying (1.2) is large. Indeed for any fixed set Ω, Ω := t·Ω verifies (1.2) for any t > 0 small. t ON THE SOBOLEV TRACE THEOREM FOR VARIABLE EXPONENT SPACES IN THE CRITICAL RANGE. 3 A more interesting and difficult task is to find local conditions on Ω ensuring (1.1). For p = 2 this was done by Adimurthy and Yadava in [1] (see also Escobar [5] for a closely related problem) by using the fact that the extremals for K¯(N,2)−1 were explicitly known since the work of Escobar [5]. In fact, in [1], theauthors proved that if theboundaryof Ω contains apoint with positive mean curvature, then (1.1) holds true. Recently Nazaret [19] found the extremals for K¯(N,p)−1 by means of mass transportation methods. These extremals are of the form Vλ,y0(y,t) = λ−Np−−1pV(y−λy0,λt), y ∈ RN−1, t > 0, with (1.3) V(y,t) = r−Np−−1p, r = (1+t)2+|y|2. From theexplicit knowledgeof theextremals onecanpcomputethevalueoftheconstant K¯(N,p) (see, for example, [8]). It holds p−1 K¯(N,p) = π1−2p p−1 p−1Γ(p2((Np−−11)))N−1 , (cid:18)N −p(cid:19) Γ( N−1 ) 2(p−1)   where Γ(x) = ∞tx−1e−tdt is the Gamma function. Using these extremals, Ferna´ndez Bonder 0 and Saintier inR[8]extended [1] by proving that (1.1)holds trueif ∂Ω contains apoint of positive mean curvature for 1 < p < (N +1)/2. See also [20] for a related result. We also refer to [22] where this question has been adressed in the case p = 1. Aslightlymoregeneralproblemcanbetreated. Namely, considerΓ ⊂ ∂Ω,Γ 6= ∂Ωa(possibly empty) closed set, and define W1,p(x) = {φ ∈ C∞(Ω¯): φ vanishes in a neighborhood of Γ}, Γ where the closure is taken in k·k −norm. This is the subspace of functions vanishing on W1,p(x)(Ω) Γ. Obviously, W1,p(x)(Ω) = W1,p(x)(Ω). In general W1,p(x)(Ω) = W1,p(x)(Ω) if and only if the ∅ Γ p(x)−capacity of Γ is 0, see [15]. The main concern of this paper is the study of the existence problem of extremals for the best constant T(p(·),r(·),Ω,Γ) defined by kvk W1,p(x)(Ω) (1.4) 0 <T(p(·),r(·),Ω,Γ) := inf . v∈W1,p(x)(Ω) kvkLr(x)(∂Ω) Γ First, employing the same ideas as in [18] we obtain some restricted conditions on the exponents p and r guarantying that the immersion W1,p(x)(Ω) ֒→ Lr(x)(∂Ω) remains compact and so the existence of an extremal for T(p(·),r(·),Ω,Γ) holds true. As in the Sobolev immersion Theorem more general conditions for the existence of extremals are needed and these are the contents of our main results. In order to state our main results, we first need to introduce some notation. The localized Sobolev trace constant T¯ is defined, for x ∈ A , as x T (1.5) T¯ = supT(p(·),r(·),Ω ,Γ ) = limT(p(·),r(·),Ω ,Γ ), x ε ε ε ε ε>0 ε→0 where Ω = Ω∩B (x) and Γ = ∂B (x)∩Ω¯. The smallest localized Sobolev trace constant is ε ε ε ε denoted by (1.6) T¯ := inf T¯ . x x∈AT 4 J.FERNA´NDEZBONDER,N.SAINTIERANDA.SILVA With these notations, our main results states that, under certain mild regularity assumptions on p and r, the following inequalities hold true T(p(·),r(·),Ω,Γ) ≤ T¯ ≤ inf K¯(N,p(x))−1. x∈AT Moreover, if the following strict inequality holds (1.7) T(p(·),r(·),Ω,Γ) < T¯, then there exists an extremal for (1.4). So a natural main concern is to provide with conditions in order for (1.7) to hold. We obtain, as in the constant exponent case, two types of conditions: local and global. Global conditions are easier to obtain. In fact, it is fairly easy to see that if Ω is contracted enough then (1.7) holds. Inorder to findlocal conditions for (1.7) to hold, amore refinedanalysis has to bemade. The idea is to find a precise test function in order to estimate T(p(·),r(·),Ω,Γ). This test function is constructedbyproperlyscalingandtruncatingtheextremalforK¯(N,p(x))−1 aroundsomepoint x ∈ A . ThisestimatewillgivelocalconditionsensuringthatT(p(·),r(·),Ω,Γ) < K¯(N,p(x))−1. T The analysis is then completed by providing with conditions that ensure T¯ = K¯(N,p(x))−1, x and requiring that T¯ = T¯ for some x ∈ A . x T Organization of the paper. The rest of the paper is organized as follows. In Section 2, we collect some preliminaries on variable exponent spaces that will be used throughout the paper. In Section 3, by applying the method developed in [18], we find conditions than ensure that the trace immersion remains compact although A 6= ∅. As we mentioned in the introduction, these T conditions are not satisfactory, so in the remaining of the paper we look for a general result that guaranty the existence of extremals. In Section 4 we revisit the proof of the Concentration– Compactness Theorem as stated in [12] to perform the corresponding adaptation for the trace inequality. In Section 5 we prove our main results, Theorem 5.4 and Theorem 5.6 that provide with general conditions for the existence of extremals. Finally, in Section 6 we provide both local and global conditions for the validity of T(p(·),r(·),Ω) < T¯. 2. Preliminaries on variable exponent Sobolev spaces In this section we review some preliminary results regarding Lebesgue and Sobolev spaces with variable exponent. All of these results and a comprehensive study of these spaces can be found in [4]. The variable exponent Lebesgue space Lp(x)(Ω) is defined by Lp(x)(Ω) = u ∈L1 (Ω): |u(x)|p(x)dx < ∞ . loc n ZΩ o This space is endowed with the norm u(x) p(x) kuk = kuk := inf λ > 0 : dx ≤ 1 Lp(x)(Ω) p(x) n ZΩ(cid:12) λ (cid:12) o (cid:12) (cid:12) We can define the variable exponent Sobolev space W1,p(x(cid:12))(Ω) b(cid:12)y W1,p(x)(Ω)= {u ∈ Lp(x)(Ω): ∂ u∈ Lp(x)(Ω) for i= 1,...,N}, i ON THE SOBOLEV TRACE THEOREM FOR VARIABLE EXPONENT SPACES IN THE CRITICAL RANGE. 5 where ∂ u= ∂u is the ith−distributional partial derivative of u. This space has a corresponding i ∂xi modular given by ρ (u) := |u|p(x)+|∇u|p(x)dx 1,p(x) Z Ω which yields the norm u kuk = kuk := inf λ > 0: ρ ≤ 1 . W1,p(x)(Ω) 1,p(x) 1,p(x) λ n (cid:16) (cid:17) o Another possible choice of norm in W1,p(x)(Ω) is kuk +k∇uk . Both norms turn out to p(x) p(x) be equivalent but we use the first one for convenience. The following result is proved in [6, 16] (see also [4], pp. 79, Lemma 3.2.20 (3.2.23)). Proposition 2.1 (H¨older-type inequality). Let f ∈ Lp(x)(Ω) and g ∈ Lq(x)(Ω). Then the following inequality holds s + s + kf(x)g(x)k ≤ + kfk kgk , Ls(x)(Ω) p q Lp(x)(Ω) Lq(x)(Ω) (cid:16)(cid:16) (cid:17) (cid:16) (cid:17) (cid:17) where 1 1 1 = + . s(x) p(x) q(x) From now on, we define the classes of exponents that we deal with. Let P(Ω) be the set of Lebesgue measurable functions p: Ω → [1,∞) and let P(∂Ω) be the set of HN−1−measurable functions r: ∂Ω → [1,∞). In order to state the trace Theorem we need to define the Lebesguespaces on ∂Ω. We assume that Ω is C1 so ∂Ω is a (N −1)−dimensional C1 immersed manifold on RN (less regularity on ∂Ω is enough for the trace Theorem to hold, but the C1 regularity is enough for our purposes). Therefore the boundary measure agrees with the (N −1)−Hausdorff measure restricted to ∂Ω. We denote this measure by dS. Then, the Lebesgue spaces on ∂Ω are defined as Lr(x)(∂Ω) := u∈ L1 (∂Ω,dS): |u(x)|r(x)dS < ∞ loc n Z∂Ω o and the corresponding (Luxemburg) norm is given by u(x) r(x) kuk = kuk := inf λ > 0: dS ≤ 1 . Lr(x)(∂Ω) r(x),∂Ω n Z∂Ω(cid:12) λ (cid:12) o (cid:12) (cid:12) Throughout this paper the following notation will be use(cid:12)d: For(cid:12) a µ−measurable function f we denote f+ := supf and f− := inff, where by sup and inf we denote the essential supremum and essential infimum respectively with respect to the measure µ. The Sobolev trace Theorem is proved in [6]. When the exponent is critical, it requires more regularity on the exponent p(x) (Lipschitz regularity is enough). This regularity can be relaxed when the exponent is strictly subcritical. It holds, Theorem 2.2. Let Ω ⊆ RN be an open bounded domain with Lipschitz boundary and let p ∈ P(Ω) be such that p ∈ W1,γ(Ω) with 1 ≤ p ≤ p+ < N < γ. Then there is a continuous − boundary trace embedding W1,p(x)(Ω)⊂ Lp∗(x)(∂Ω). Theorem 2.3. Let Ω ⊂ RN be an open bounded domain with Lipschitz boundary. Suppose that p ∈ C0(Ω¯) and 1 <p− ≤ p+ < N. If r ∈ P(∂Ω) is uniformly subcritical then the boundary trace embedding W1,p(x)(Ω) → Lr(x)(∂Ω) is compact. 6 J.FERNA´NDEZBONDER,N.SAINTIERANDA.SILVA Corollary 2.4. Let Ω ⊂ RN be an open bounded domain with Lipschitz boundary. Suppose that p ∈ C0(Ω¯) and 1 < p ≤ p < N. If r ∈ C0(∂Ω) satifies the condition − + 1 ≤ r(x)< p (x) x ∈ ∂Ω ∗ then there is a compact boundary trace embedding W1,p(x)(Ω)→ Lr(x)(∂Ω) The following proposition, also proved in [16], will be most useful (see also [4], Chapter 2, Section 1). Proposition 2.5. Set ρ(u) := Ω|u(x)|p(x)dx. For u ∈ Lp(x)(Ω) and {uk}k∈N ⊂ Lp(x)(Ω), we have R u (2.1) u 6= 0⇒ kuk = λ ⇔ ρ( ) = 1 . Lp(x)(Ω) λ (cid:16) (cid:17) (2.2) kuk < 1(= 1;> 1) ⇔ ρ(u) < 1(= 1;> 1). Lp(x)(Ω) p− p+ (2.3) kuk > 1 ⇒ kuk ≤ ρ(u) ≤ kuk . Lp(x)(Ω) Lp(x)(Ω) Lp(x)(Ω) p+ p− (2.4) kuk < 1 ⇒ kuk ≤ ρ(u) ≤ kuk . Lp(x)(Ω) Lp(x)(Ω) Lp(x)(Ω) (2.5) lim ku k = 0 ⇔ lim ρ(u )= 0. k Lp(x)(Ω) k k→∞ k→∞ (2.6) lim ku k = ∞ ⇔ lim ρ(u ) = ∞. k Lp(x)(Ω) k k→∞ k→∞ For much more on these spaces, we refer to [4]. 3. Compact case In this section we find conditions on the exponents p ∈ P(Ω) and r ∈ P(∂Ω) that imply that the immersion W1,p(x)(Ω) ֒→ Lr(x)(∂Ω) remains compact. Therefore, in this case, the existence of extremals follows directly by minimization. Roughlyspeaking,theseconditionsrequirethecriticalsettobesmall, andalsoastrictcontrol on how the exponent r reaches the critical one when one is approaching the critical set A . For T the Sobolev immersion W1,p(x)(Ω) ֒→ Lq(x)(Ω), this result was obtained in [18]. Following the 0 same ideas we can prove a similar result for the trace immersion. First,wedefinetheupperMinkowskycontentforsetscontainedin∂Ω. Wesaythatacompact set K ⊂ ∂Ω has finite (N −1−s)−boundary dimensional upper Minkowsky content if there exists a constant C > 0 such that HN−1(K(r)∩∂Ω) ≤ Crs, for all r > 0, where K(r)= {x ∈ RN: dist(x,K) < r}. The result is the following: Theorem 3.1. Let ϕ: [r−1,∞) → (0,∞) be a continuous function such that: ϕ(r)/lnr is 0 nonincreasing in [r−1,∞) for some r ∈ (0,e−1) and ϕ(r) → ∞ as r → ∞. Let K ⊂ ∂Ω be 0 0 a compact set whose (N −1−s)−boundary dimensional upper Minkowski content is finite for some s with 0 < s≤ N −1. Let p ∈ P(Ω) and r ∈ P(∂Ω) be such that p+ < N and r(x) ≤ p (x). Assume that r(x) is ∗ subcritical outside a neighborhood of K, i.e. inf (p (x)−r(x)) > 0. Moreover, assume ∂Ω\K(r0) ∗ ON THE SOBOLEV TRACE THEOREM FOR VARIABLE EXPONENT SPACES IN THE CRITICAL RANGE. 7 that r(x) reaches p (x) in K at the following rate ∗ ϕ( 1 ) dist(x,K) r(x) ≤ p (x)− for almost every x ∈ K(r )∩∂Ω. ∗ ln( 1 ) 0 dist(x,K) Then the embedding W1,p(x)(Ω) ֒→ Lr(x)(∂Ω) is compact. Proof. Let us prove that (3.1) lim sup |v(x)|r(x)dS: v ∈ W1,p(x)(Ω) and kvk ≤ 1 = 0. ε→0+ nZK(ε)∩∂Ω W1,p(x)(Ω) o First, we take β such that 0 < β < s/p+ and ε > 0 such that ε−1 > r−1 and ϕ(1) ≥ 1. For ∗ 0 ε each n∈ N we consider η = ε−βn. We choose x ∈ (K(εn)\K(εn+1))∩∂Ω, then , we have n −ϕ(cid:18)dist(1x,K)(cid:19) −ϕ(εn1+1) ηnr(x)−p∗(x) ≤ ηn ln(cid:18)dist(1x,K)(cid:19) ≤ ηn ln(εn1+1) = ε−nβ+n1ϕ(cid:16)εn1+1(cid:17) = An. On the other hand, we know that H(K(r)∩∂Ω) ≤Crs and we can estimate the following term ηr(x)dS ≤ ηp+∗ dS ≤ Cεn(s−βp+∗) n n Z(K(εn)\K(εn+1))∩∂Ω ZK(εn)∩∂Ω Now, we have |v(x)|r(x)dS Z(K(εn)\K(εn+1))∩∂Ω |v(x)| p∗(x)−r(x) ≤ |v(x)|r(x) dS + ηr(x)dS Z(K(εn)\K(εn+1))∩∂Ω (cid:18) ηn (cid:19) Z(K(εn)\K(εn+1))∩∂Ω n ≤ A |v(x)|p∗(x)dS +Cεn(s−βp+∗) n Z(K(εn)\K(εn+1))∩∂Ω for each n ∈ N, we obtain 0 ∞ |v(x)|r(x)dS = |v(x)|r(x)dS ZK(εn0)∩∂Ω nX=n0Z(K(εn)\K(εn+1))∩∂Ω ∞ ≤ (sup A ) |v(x)|p∗(x)dS +C εn(s−βp+∗) n n≥n0 ZK(εn0)∩∂Ω nX=n0 Using that kvk ≤ Ckvk and that (s−βp+)> 0, we can conclude (3.1). p∗,∂Ω 1,p ∗ Finally, let {vn}n∈N ⊂ W1,p(x)(Ω) and v ∈ W1,p(x)(Ω) be such that v ⇀ v weakly in W1,p(x)(Ω). n Then, v ⇀ v weakly in Lr(x)(∂Ω), n v → v strongly in Ls(x)(∂Ω) for every s such that inf(p (x)−s(x))> 0, n ∗ ∂Ω 8 J.FERNA´NDEZBONDER,N.SAINTIERANDA.SILVA therefore v → v in Lr(x)(∂Ω\K(ε)) for each ε> 0 small. Hence, n limsup |v (x)−v(x)|r(x)dS =limsup |v (x)−v(x)|r(x)dS n n n→∞ Z∂Ω n→∞ (cid:16)ZK(ε)∩∂Ω + |v (x)−v(x)|r(x)dS n Z∂Ω\K(ε) (cid:17) ≤sup |v (x)−v(x)|r(x)dS n n∈NZK(ε)∩∂Ω So, by (3.1), we conclude the desired result. (cid:3) Now it is straightforward to derive, analogous to Corollary 3.5 in [18], Corollary 3.2. Let p ∈P(Ω) be such that p+ < N and let r ∈ P(∂Ω). Suppose that there exist x ∈ Ω, C > 0, n ∈ N, r > 0 such that inf (p (x)−r(x)) > 0 and r(x) ≤ p (x)− 0 0 ∂Ω\Br0(x0) ∗ ∗ lnn( 1 ) c |x−x0| for almost every x ∈ ∂Ω∩B (x ). Then the embedding W1,p(x)(Ω) ֒→ Lr(x)(∂Ω) is ln( 1 ) r0 0 |x−x0| compact. 4. The concentration–compactness principle for the Sobolev trace immersion This section is devoted to the extension of the CCP to the trace immersion. Let r ∈ P(∂Ω) be a continuous critical exponent in the sense that A := {x ∈∂Ω: r(x) =p (x)} =6 ∅. T ∗ 1,p(x) We define the Sobolev trace constant in W (Ω) as Γ kvk kvk 1,p(x) 1,p(x) T(p(·),r(·),Ω,Γ) := inf = inf v∈W1,p(x)(Ω) kvkr(x),∂Ω v∈W1,p(x)(Ω) kvkr(x),∂Ω\Γ Γ Γ More precisely, we prove Theorem 4.1. Let {un}n∈N ⊂ W1,p(x)(Ω) be a sequence such that un ⇀ u weakly in W1,p(x)(Ω). Then there exists a countable set I, positive numbers {µ } and {ν } and points {x } ⊂ i i∈I i i∈I i i∈I A ⊂ ∂Ω such that T (4.1) |u |r(x)dS ⇀ ν = |u|r(x)dS + ν δ weakly-* in the sense of measures, n i xi Xi∈I (4.2) |∇u |p(x)dx ⇀ µ ≥ |∇u|p(x)dx+ µ δ weakly-* in the sense of measures, n i xi Xi∈I 1 1 (4.3) T¯ νr(xi) ≤ µp(xi), xi i i where T¯ = sup T(p(·),q(·),Ω ,Γ ) is the localized Sobolev trace constant where xi ε>0 ε,i ε,i Ω = Ω∩B (x ) and Γ := ∂B (x )∩Ω. ε,i ε i ε,i ε i Proof. The proof is very similar to the one for the Sobolev immersion Theorem, see [11], so we only make a sketch stressing the differences between the two cases. As in [12, Theorem 1.1] it is enough to consider the case where u ⇀ 0 weakly in W1,p(x)(Ω). n ON THE SOBOLEV TRACE THEOREM FOR VARIABLE EXPONENT SPACES IN THE CRITICAL RANGE. 9 Take φ ∈ C∞(Ω¯). According to Theorem 2.2 we have (4.4) T(p(·),q(·),Ω)kφu k ≤ kφu k . j r(x) j 1,p(x) We have that kφu k ≤ C(k∇(φu )k +kφu k ) j 1,p(x) j p(x) j p(x) On the other hand, | k∇(φu )k −kφ∇u k | ≤ ku ∇φk . j p(x) j p(x) j p(x) Then, as u ⇀ 0, we observe that the right hand side of the inequality converges to 0. In fact, n we can assume that ρ (u) < 1, then p(x) ku ∇φk ≤ (k∇φk +1)p+ku k j p(x) ∞ j p(x) ≤ (k∇φk +1)p+ρ (u )1/p− → 0 ∞ p(x) j We the same argument, we obtain that kφu k → 0 j p(x) Finally, if we take the limit for j → ∞ in (4.4), we arrive at (4.5) T(p(·),r(·),Ω)kφk ≤ kφk , Lrν(x)(∂Ω) Lpµ(x)(Ω) for every φ ∈C∞(Ω¯). Observe that if φ∈ C∞(RN) and U ⊂ RN is any open set containing the c support of φ, the constant in (4.5) can be replaced by T(p(·),q(·),Ω∩U,∂U ∩Ω). Now, the exact same proof of [12, Theorem 1.1] implies that the points {x } must belong i i∈I to the critical set A . T Let φ∈ C∞(RN) be such that 0≤ φ ≤ 1, φ(0) = 1 and supp(φ)⊂ B (0). Now, for each i ∈I c 1 and ε> 0, we denote φ (x) := φ((x−x )/ε). ε,i i From (4.5) and the subsequent remark we obtain T(p(·),r(·),Ω ,Γ )kφ k ≤ kφ k . ε,i ε,i ε,i Lrν(x)(∂Ω∩Bε(xi)) ε,i Lpµ(x)(Ω∩Bε(xi)) As in [12], we have ρ (φ ) := |φ |r(x)dν ν i0,ε Z i0,ε ∂Ω∩Bε(xi0) = |φ |r(x)|u|r(x)dS + ν φ (x )r(xi) Z i0,ε i i0,ε i ∂Ω∩Bε(xi0) Xi∈I ≥ ν . i0 From now on, we will denote r+ := sup r(x), r− := inf r(x), i,ε i,ε ∂Ω∩Bε(xi) ∂Ω∩Bε(xi) p+ := sup p(x), p− := inf p(x). i,ε i,ε Ω∩Bε(xi) Ω∩Bε(xi) If ρ (φ ) < 1 then ν i0,ε kφi0,εkLνr(x)(∂Ω∩Bε(xi0)) ≥ ρν(φi0,ε)1/ri−,ε ≥ νi10/ri−,ε. 10 J.FERNA´NDEZBONDER,N.SAINTIERANDA.SILVA Analogously, if ρ (φ ) > 1 then ν i0,ε 1/r+ kφ k ≥ ν i,ε. i0,ε Lrν(x)(∂Ω∩Bε(xi0)) i0 Therefore, 1 1 r+ r− T(p(·),r(·),Ω ,Γ )min ν i,ε,ν i,ε ≤ kφ k . ε,i ε,i n i i o i,ε Lpµ(x)(Ω∩Bε(xi)) On the other hand, |φ |p(x)dµ ≤ µ(Ω∩B (x )) i,ε ε i Z Ω∩Bε(xi) hence 1 1 kφi,εkLp(x)(Ω∩Bε(xi)) ≤ max ρµ(φi,ε)p+i,ε,ρµ(φi,ε)p−i,ε n o 1 1 ≤ max µ(Ω∩Bε(xi))p+i,ε,µ(Ω∩Bε(xi))p−i,ε , n o so we obtain, T(p(·),r(·),Ωε,i,Γε,i)min νiri+1,ε,νiri−1,ε ≤ max µ(Ω∩Bε(xi))p+i1,ε,µ(Ω∩Bε(xi))p−i1,ε . n o n o As p and r are continuous functions and as r(x ) = p (x ), letting ε → 0, we get i ∗ i T¯ ν1/p∗(xi) ≤ µ1/p(xi), xi i i where µ := lim µ(Ω∩B (x )). i ε→0 ε i The proof is now complete. (cid:3) 5. Non-compact case In this section we parallel the results for the Sobolev immersion Theorem obtained in [11], to the Sobolev trace Theorem. In that spirit, the result we obtain states that if the Sobolev trace constant is strictly smaller that the smallest localized Sobolev trace constant in the critical set A , then there exists an T extremal for the trace inequality. Then,theobjective willbetofindconditions onp(x),r(x) andΩinorderto ensurethat strict inequality. We find global and local conditions. Asin[11], global conditionsareeasily obtainedandthey say thatif thesurfacemeasureofthe boundaryis larger than the volume of the domain, then the strict inequality holds and therefore an extremal for the trace inequality exists. Once again, local conditions are more difficult to find. In this case, the geometry of the domain comes into play. We begin with a lemma that gives a bound for the constant T(p(·),r(·),Ω,Γ). Lemma 5.1. Assume that the exponents p ∈ P(Ω) and r ∈ P(∂Ω) are continuous functions with modulus of continuity ρ such that ln(λ)ρ(λ) → 0 as λ → 0+.