PREPUBLICACIONES DEL DEPARTAMENTO DE MATEMÁTICA APLICADA UNIVERSIDAD COMPLUTENSE DE MADRID MA-UCM

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1 PREPUBLICACIONES DEL DEPARTAMENTO DE MATEMÁTICA APLICADA UNIVERSIDAD COMPLUTENSE DE MADRID MA-UCM THE BEST SOBOLEV TRACE CONSTANT AS LIMIT OF THE USUAL SOBOLEV CONSTANT FOR SMALL STRIPS NEAR THE BOUNDARY J.M. Arrieta, A. Rodríguez-Bernal and J.Rossi Marzo

2 THE BEST SOBOLEV TRACE CONSTANT AS LIMIT OF THE USUAL SOBOLEV CONSTANT FOR SMALL STRIPS NEAR THE BOUNDARY JOSÉ M. ARRIETA, ANÍBAL RODRÍGUEZ-BERNAL AND JULIO D. ROSSI Abstract. In this paper we prove that the best constant in the Sobolev trace embedding H (Ω) L q () in a bounded smooth domain can be obtained as the limit as 0 of the best constant of the usual Sobolev embedding H (Ω) L q (ω, dx/) where ω = {x Ω : dist(x, ) < } is a small neighborhood of the boundary. We also analyze symmetry properties of extremals of this last embedding when Ω is a ball.. Introduction. The main goal of this article is to obtain the best Sobolev trace constant for a given domain as the limit of the usual Sobolev constant in small strips near the boundary of the domain when the width of the strip goes to zero. Sobolev inequalities have been studied by many authors and is by now a classical subject. It at least goes back to [2], for more references see [5]. Relevant for the study of boundary value problems for differential operators is the Sobolev trace inequality that has been intensively studied, see for example, [3], [7], [8], [9], [0]. In this paper we consider the best Sobolev trace constant. Given a bounded smooth domain Ω R N, we deal with the best constant of the Sobolev trace embedding H (Ω) L q (). For every critical or subcritical exponent, q 2 = 2(N )/(N 2), we have the Sobolev trace inequality: there exists a constant C such that ( 2/q C v ds) q ( v 2 + v 2 ) dx, Ω Key words and phrases. Nonlinear boundary conditions, Sobolev trace embedding, symmetry of extremals Mathematics Subject Classification. 35J65, 46E35.

3 2 J.M. ARRIETA, A. RODRIGUEZ-BERNAL AND J.D. ROSSI for all v H (Ω). The best Sobolev trace constant is the largest C such that the above inequality holds, that is, v 2 + v 2 dx Ω (.) T q = inf ( ) v H (Ω)\H0 (Ω) 2/q. v q ds For subcritical exponents, q < 2, the embedding is compact, so we have existence of extremals, i.e. functions where the infimum is attained. These extremals can be taken strictly positive in Ω and smooth up to the boundary. If we normalize the extremals with (.2) u q ds =, it follows that they are weak solutions of the following problem u + u = 0 in Ω, (.3) u ν = T q u q 2 u on, where ν is the unit outward normal vector. In the special case q = 2 (.3) is a linear eigenvalue problem of Steklov type, see [7]. In the rest of this article we will assume that the extremals are normalized according to (.2). As we have mentioned, we want to see how the best trace constant, T q, can be obtained as the limit of the usual Sobolev constant for some subdomains. To this end, let us consider the subset of Ω ω = {x Ω : dist(x, ) < }. Notice that this set has measure ω for small values of. For sufficiently small σ 0 we can define the parallel interior boundary Γ σ = {y σν(y), y }, where ν(y) denotes the outward unitary normal at y. Note that Γ 0 =. Then, we can also look at the set ω as the neighborhood of Γ 0 defined by ω = {x = y σν(y), y, σ [0, )} = 0 σ< for sufficiently small, say 0 < < 0. We also denote by Ω δ = {x Ω : dist(x, ) > δ} and for δ small we have that δ = Γ δ. Let us consider the usual Sobolev embedding associated to the set ω, that is, ( H (Ω) L q ω, dx ). Γ σ

4 SOBOLEV TRACE CONSTANT 3 We have normalized the size of ω by taking dx/ as measure in ω. In this case the embedding is continuous for exponents q such that q 2 = 2N/(N 2). Note that 2 = 2N/(N 2) is larger than 2 = 2(N )/(N 2). The best constant associated to this embedding is given by v 2 + v 2 dx (.4) S q () = inf v H (Ω) Ω ( ) 2/q. v q dx ω For q < 2, by compactness, the infimum is attained. The extremals, normalized by (.5) u q dx =, ω are weak solutions of u + u = S q() χ ω (x) u q 2 u in Ω, (.6) u ν = 0 on, where χ ω denotes the characteristic function. Our main result is the following: Theorem. Let T q and S q () be the best Sobolev constants given by (.) and (.4). () For critical or subcritical q, q 2 = 2(N )/(N 2), we have (.7) lim S q () = T q. Moreover, for subcritical q, q < 2 = 2(N )/(N 2), the extremals of S q () normalized according to (.5) converge strongly (along subsequences) in H (Ω) and in C β (Ω), for some β > 0, to an extremal of (.), lim u = u 0, strongly in H (Ω) and in C β (Ω). In the critical case, q = 2 = 2(N )/(N 2), the extremals of S q () converge weakly (along subsequences) in H (Ω) to a limit, u 0, that is a weak solution of (.3). This convergence is strong in H (Ω) if and only if the limit verifies uq 0 = and in this case u 0 is an extremal for T 2.

5 4 J.M. ARRIETA, A. RODRIGUEZ-BERNAL AND J.D. ROSSI (2) For supercritical q, 2 = 2(N )/(N 2) < q < 2 = 2N/(N 2), we have (.8) lim S q () = 0. A reference closely related to this work is [] where the authors consider concentrated reactions near the boundary in an elliptic problem. They prove that the solutions converge to a solution of a problem with a nonhomogeneous flux condition at the boundary. Our results can be viewed as a complement of the results of [] since here we deal with (nonlinear) eigenvalue problems when the reactions are concentrated near the boundary (see the right hand side in (.6)). Next, we look at the symmetry for extremals of (.4) in the special case when Ω is a ball, Ω = B(0, R). In this case we prove the following result. Theorem 2. Let S q () the best Sobolev constant given by (.4) with Ω = B(0, R). () For q 2 and for every R, > 0, the extremals of (.4) in a ball are radial functions that do not change sign. In particular, there exists a unique non negative extremal of (.4) satisfying (.5). (2) For 2 < q < 2 = 2(N )/(N 2), there exist 0 < R 0 R < such that: (2.) for 0 < R R 0 and small (possibly depending on R) the extremals of (.4) are radial. (2.2) for R R and small (possibly depending on R) the extremals of (.4) are not radial. 2. Proof of Theorem This section is devoted to the proof of Theorem. First, we prove that the Sobolev trace constant is continuous as a function of the domain. We believe that this result has independent interest by itself. Lemma 2.. Let Ω δ = {x Ω : dist(x, ) > δ}. Then the function is continuous at δ = 0. δ T q (Ω δ ), Proof. Consider a fixed 0 > 0 small enough. For all 0 < δ < 0, let us consider a smooth increasing function ψ δ such that ψ δ (0) = δ,

6 SOBOLEV TRACE CONSTANT 5 ψ δ (s) = s for all s 0 and ψ δ (s) s as δ 0 in C ([0, )). Now we take the diffeomorphism A δ : Ω Ω δ, { y ψ δ (s)ν(y) for x = y sν(y) ω, s (0, ), A δ (x) = x for x Ω \ ω. which is also a diffeomorphism when restricted to the boundary, A δ : δ. This diffeomorphism has bounded derivatives and moreover (2.) lim δ 0 DA δ (x) I = 0, uniformly in Ω. Here I M n n is the identity matrix. Therefore, we can change variables with, u(x) = v(a δ (x)), for x Ω or x. This induces a map, that we denote the same A δ : H (Ω) H (Ω δ ) which is a diffeomorphism. Moreover, we have that the following diagram is commutative (2.2) H (Ω) L q () A δ A δ H (Ω δ ) L q ( δ ). Therefore, from (2.), we obtain C (δ) u 2 + u 2 dx v 2 + v 2 dx C 2 (δ) Ω Ω δ where C i (δ) as δ 0. In a similar way, we get (2.3) C (δ) u q ds v q ds C 2 (δ) δ with C i (δ) as δ 0. Ω u 2 + u 2 dx, u q ds, From the previous inequalities we obtain that there exist two constants K, K 2 such that K i (δ) as δ 0 and K (δ) T q (Ω) T q (Ω δ ) K 2 (δ) T q (Ω). The desired continuity is proved.

7 6 J.M. ARRIETA, A. RODRIGUEZ-BERNAL AND J.D. ROSSI The next result shows that the traces on δ also behave continuously as δ 0. In order to do this, we first figure out a device that allows to compare traces taken on different surfaces close to the boundary of Ω. For this observe that, for any q 2 we can define the mapping γ δ : H (Ω) L q ( δ ) L q (). Here the first arrow denotes traces and the second one denotes the diffeomorphism induced by A δ as in (2.2). Then, we have the following result, which in particular complements some results in []. Lemma 2.2. Denoting by γ the trace operator on, we have lim γ δ = γ in L q () δ 0 on compact sets of H (Ω) if q = 2 or in L(H (Ω), L q ()) if q < 2. In particular, for q 2, if u is a bounded sequence in H (Ω), then ω u q is also bounded. Moreover, if u u 0 strongly in H (Ω) and q 2, then (2.4) u q ds u 0 q ds δ() as δ() 0 and (2.5) as 0. u q dx u 0 q ds ω Proof. If q 2 and u is a bounded sequence in H (Ω), we write u q dx = u q ds dδ ω 0 δ 0 T q (Ω δ ) q 2 u q H (Ω δ ) dδ sup [T q (Ω δ ) q 2 ] u q H (Ω) δ [0,] which is bounded using Lemma 2. and the fact that the sequence u is bounded in H (Ω). Note that if q < 2 there exists some 0 < s <, such that γ δ : H (Ω) H s (Ω) L q ( δ ) L q (). In a similar fashion, if q = 2, we take s =.

8 SOBOLEV TRACE CONSTANT 7 For any fixed u H s (Ω), from (2.3), we have that these operators converge to the usual trace on, that is Moreover, we have lim γ δ(u) = γ(u). δ 0 γ δ L(H s (Ω),L q ()) C, uniformly on δ. Hence, from the Banach Alouglu Bourbaki lemma, we get on compact sets of H s (Ω). lim γ δ = γ δ 0 In addition, if u u 0 strongly in H (Ω) γ (u ) q ds = u 0 q ds lim which combined with (2.3) gives (2.4). On the other hand, to obtain (2.5) we write u q dx = u q ds dδ. ω 0 δ Since for every δ <, δ u q and u 0 q are uniformly close, we get (2.5). Remark 2.3. The only property that we have actually used in the proof of the previous results is (2.). Therefore both lemmas above remain true for any family of domains Ω δ such that there exists a diffeomorphism A δ : Ω Ω δ with A δ : δ such that (2.) holds. Also note that in Lemma 2.2 the conclusions remain true for q < 2 under the weaker assumption of convergence in H s (Ω) for s < but close enough to. Proof of Theorem. We first prove (.7) for critical or subcritical exponents, i.e. q 2 = 2(N )/(N 2). Given k > 0, let us take a regular function u k such that u k 2 + u 2 k dx T q + k Ω ( ) 2/q. u q k dx ω

9 8 J.M. ARRIETA, A. RODRIGUEZ-BERNAL AND J.D. ROSSI By the regularity of u k, from Lemma 2.2 (see also []), we have, for a fixed k, lim u q k dx = lim u q ω k d x ds = u q k d x. 0 Γ s Therefore, using u k as test in (.4) and taking limits we get Letting k we obtain lim sup S q () T q + k. (2.6) lim sup S q () T q. Now let us prove that for q 2 we have (2.7) lim inf S q () T q. For this, note that for u H (Ω) we get, using the restriction to Ω δ, ( ) 2/q u q ds δ T q (Ω δ ) u 2 H (Ω δ ) T q (Ω δ ) u 2 H (Ω). Integrating for δ (0, ) we obtain u q dx = u q dδ ω 0 δ Thus, we have obtained ( (2.8) 0 ( 0 ) 2/q dδ S (T q (Ω δ )) q/2 q (). This fact, together with the continuity of the map proved in Lemma 2., gives (2.7). From (2.6) and (2.7) we obtain as we wanted to prove. δ T q (Ω δ ), lim S q() = T q, ) dδ u q (T q (Ω δ )) q/2 H (Ω). Now we turn our attention to the convergence of extremals in the subcritical case q < 2. To prove this fact, let us consider u an extremal of S q () normalized by (2.9) ω u q dx =.

10 Hence we have, for small, using (2.6), SOBOLEV TRACE CONSTANT 9 u 2 H (Ω) = S q() T q +. Therefore the sequence u is bounded in H (Ω) and we can extract a subsequence (that we still denote u ) such that (2.0) Now we claim that, (2.) u u 0 weakly in H (Ω), u u 0 strongly in L 2 (Ω), u u 0 strongly in H s (Ω), s <, u u 0 strongly in L q (), u u 0 a.e. in Ω. To prove this, note that as we have = u q dx = ω u 0 q ds =. 0 δ u q ds dδ, from the integral mean value theorem, there exists 0 δ() such that δ u q ds =. Now, from the convergence of u to u 0 in H s (Ω), valid for 0 < s <, we conclude that u 0 q ds = see the Remark after Lemma 2.2. This finishes the proof of the claim. With this in mind, we have u u 2 0 dx Ω T q ( ) 2/q u 0 2 H (Ω) u 0 q ds lim sup u 2 H (Ω) lim inf = lim sup S q () = T q. u 2 H (Ω) Therefore lim u H (Ω) = u 0 H (Ω). In particular, the convergence of the norms implies that the extremals of S q () normalized according to (2.9) converge strongly in H (Ω) to

11 0 J.M. ARRIETA, A. RODRIGUEZ-BERNAL AND J.D. ROSSI an extremal of (.), which satisfies (2.). lim u = u 0, strongly in H (Ω) Now, let us prove that we have convergence in C β (Ω), for some β > 0. To this end we will use some results from [] that describe the behavior of solutions of linear elliptic equations with concentrated potentials. Denote by V (x) = S q ()u q 2 so that u is a solution of the problem u + u = χ ω V u in Ω, u ν = 0 on. First, remark that as q is subcritical we can choose r > N such that V ω r dx = S q() r u (q 2)r dx C, ω with C independent of. Indeed, as u is uniformly bounded in H (Ω) we have from Lemma 2.2, that for any θ 2(N )/(N 2), u θ dx C. ω Now, just write θ = (q 2)r and use the fact that q < 2(N )/(N 2) (this implies (q 2) < 2/(N 2)) to obtain that for some r > N we have (q 2)r 2(N )/(N 2). Moreover, since S q () T q, u u 0 in H (Ω) and q is subcritical, we have that V φ dx V 0 φ ds ω for any smooth function φ, where V 0 (x) = T q u q 2 0 (x). Hence, u 0 satisfies u 0 + u 0 = 0 in Ω, u 0 ν = V 0u 0 on. With all this at hand, we can apply Theorem 3. and Corollary 3.2 from [], that guarantee the convergence in the Hölder norm C β (Ω), for some β > 0. In the critical case q = 2 we also obtain a uniform bound in H (Ω) for the extremals u of S q (). Therefore we can extract a subsequence such that (2.0) holds. Passing to the limit in the weak form of (.6) we get that the limit u 0 is a weak solution of (.3). However, due to

12 SOBOLEV TRACE CONSTANT the lack of compactness, we cannot ensure that u 0 verifies u 0 q = in this case. To finish the proof of the theorem it remains to show (.8) in the supercritical case 2 = 2(N )/(N 2) < q < 2 = 2N/(N 2). To see this fact assume that 0 and consider u(x) = x λ. Where we choose λ such that u H (Ω), i.e. λ < (N 2)/2. Now we choose λ = λ(q) such that u q ds = +, that is, λ (N )/q, which is possible since q > 2. We observe that with this choice, we have lim u q dx = +. ω The proof is finished. Remark 2.4. Observe that in the critical case, using a sequence of minimizers and subsequences if necessary we have u u 0 weakly in H (Ω) and S (q) T q. Also, we have u 0 2 H (Ω) and lim inf u 2 H (Ω) T q lim sup u 2 H (Ω) u u 2 0dx Ω ( ) 2/q. u 0 q ds = lim sup S q () = T q Hence if u 0 is a minimizer, then u 0 q ds. Conversely, if u 0 q ds then the argument above shows that this integral is actually equal to and u 0 is a minimizer. Moreover in such a case, we get the convergence of the H (Ω) norms and hence the strong convergence in this space. Thus, u 0 is a minimizer if and only if u 0 q ds = which in turn is equivalent to the strong convergence. Also, in the critical case it may happen then that one has (.5) and u 0 q ds <.

13 2 J.M. ARRIETA, A. RODRIGUEZ-BERNAL AND J.D. ROSSI 3. Proof of Theorem 2 We divide the proof of Theorem 2 in several lemmas. section we take Ω = B(0, R), except in the next result. Along this Lemma 3.. Let Ω be arbitrary. Then for any q 2 and any > 0 every extremal is of constant sign. Moreover, there exists a unique positive extremal of (.4), normalized according to (.5). Proof. Note that non negative extremals of (.4) are indeed positive solutions of (.6), i.e. they satisfy, when normalized as in (.5), u = f(x, u) = a(x)u ρ u in Ω, (3.) u ν = 0 on, where a(x) = Sq() χ ω (x) 0 and ρ = q. Also, note that from (.4), non negative extremals exists, since the absolute value of an extremal is an extremal. Now, we use an argument from [8]; see also [5] and [6]. Note that if q < 2 then ρ <. Hence, if x Ω \ ω we have f(x, u) = u C(x)u + D(x) if we take C(x) = and D(x) = 0. On the other hand, if x ω Young s inequality yields for sufficiently small δ, f(x, u) ( δ ) [ ] Sq () ρ u + β δ ρ for some constant β > 0 and we can take C(x) = δ and [ ] Sq () ρ D(x) = β. δ ρ In summary C(x) = δχ ω (x), and we have, for u > 0 and x Ω, D(x) = β f(x, u) C(x)u + D(x). [ ] Sq () ρ δ ρ χω (x) Note that for sufficiently small δ, the semigroup generated by + C(x) in Ω with Neumann boundary conditions decays exponentially. Then since D L (Ω), we get from [8] and [5], that there exist a solution of (3.), which is maximal in the sense of pointwise ordering. In particular it is nonzero since it bounds above in a pointwise sense any normalized positive extremal.

14 SOBOLEV TRACE CONSTANT 3 Now, the proof concludes by showing that in fact (3.) has a unique solution, which follows from the fact that f(x, u) = a(x) u u ρ is nonicreasing for u > 0 and strictly decreasing on a set of positive measure. Indeed, let ϕ be the maximal positive solution of (3.) and 0 < ψ ϕ any other solution. Then, multiplying the equation satisfied by ϕ by ψ and the one for ψ by ϕ, substracting and integrating by parts in Ω, we have 0 = Ω f(x, ϕ) ϕ ϕψ Ω ( f(x, ψ) f(x, ϕ) ϕψ = ψ Ω ϕ f(x,ψ) ψ Now, since ψ ϕ we have that f(x,ϕ) ϕ set of positive measure. Therefore, we must have ψ 0. ) f(x, ψ) ϕψ. ψ 0 and is non zero in a When q = 2 the conclusion of the lemma follows easily since the first eigenvalue of the elliptic problem (.6) is simple, [4]. Therefore there exists a unique positive eigenfunction such that (.5) holds. With this, if Ω = B(0, R), we get the following result, which actually proves the first part of Theorem 2. Corollary 3.2. For every q 2 and every R, > 0 every extremal of (.4) is radial and does not change sign in Ω. Proof. Note that in any case q < 2 or q = 2, the absolute value of an extremal is also an extremal. Therefore, the absolute value is a nonnegative extremal and must be then coincide with the unique positive extremal. This one, in turn, must be radial, since, by uniqueness, it must coincide with any rotation of it. The following lemma proves (2.) in Theorem 2. Lemma 3.3. For 2 < q < 2 = 2(N )/(N 2) there exists R such that for every R > R there exists 0 such that the extremals (.4) are not radial for < 0. Proof. The results of [6] imply that in this case the extremals of the best Sobolev trace constant T q (B(0, R)) are not radial (since they develop a concentration phenomena). Since the extremals for S q () converge to the extremals of T q (B(0, R)) as 0 they cannot be radial for small enough (possibly depending on R). Now we finish the proof of Theorem 2.

15 4 J.M. ARRIETA, A. RODRIGUEZ-BERNAL AND J.D. ROSSI Lemma 3.4. For 2 < q < 2 = 2(N )/(N 2) there exists R 0 such that for every R R 0 there exists 0 such that there exists a radial extremal of (.4) for < 0. Proof. First, let us choose R 0 in such a way that for any R < R 0 the problem u + R 2 u = 0 in B(0, ), (3.2) u ν = T q(r) R2 u q on B(0, ), R β has a unique positive solution close to u 0 normalized with the usual constraint B(0,) uq =, see [8]. Here β = qn 2N + 2. q Observe that the above problem is just (.3) (together with (.2)) rescaled from the ball of radius R to the ball of radius one. Also note that, from the results of [9], we have T q (R) lim = R 0 R β B(0, ) B(0, ) 2/q. Moreover, we can assume (taking R 0 smaller if necessary) that for R < R 0 the linearized of (3.2) is invertible. This can be obtained since for small R there is a unique solution to (3.2) with B(0,) uq = and the linearized problem is invertible at R = 0, u = and then B(0,) /q invertible at (R, u R ) for small R (see [8] for the details). Now we want to use the implicit function theorem in (.4). To this end, let us rescale (.6) to the unit ball defining v(x) = R α u(rx) where u is the solution of (.6) satisfying (.5). If α = (N )/q, we have that v satisfies (3.3) v q dx =, R,R where,r = B(0, ) \ B(0, R ) and also v + R 2 v = R 2 S q () R β R χ,r(x) v q in B(0, ), v = 0 ν on B(0, ),

16 SOBOLEV TRACE CONSTANT 5 where χ,r (x) is the characteristic function of,r. Let { } S = v H (B(0, )); v q ds =. B(0,) If we multiply v by an adequate constant µ in order to have w = µv S, we have µ = ( B(0,) vq ) q and we are left with a solution of w + R 2 w = R 2 Ã() (3.4) R R χ,r(x)w q in B(0, ), β w = 0 on B(0, ). ν Here ( ) 2/q Ã() = S q () v q ds B(0,) where the integral term also depends on through v. From (.7) and the convergence of the extremals in Theorem, we get, using (3.3) and Lemma 2.2, that Ã() T q as 0. Let us consider the functional given by F (w, )(φ) = F : S [0, 0 ] (H (B(0, ))), B(0,) R2 Ã() R R β w φ dx + R 2 B(0,) B(0,)\B(0, R ) wφ dx w q φ dx. This functional is C with respect to w S (since q > 2). Remark that we are looking for pairs (w, ) that are solutions of F (w, ) = 0 (these are weak solutions of (3.4)). To apply the implicit function theorem we need to compute F (u, 0). w First, let us compute the derivative F (w, )(φ)(χ) = χ φ dx + R 2 χφ dx w B(0,) B(0,) R2 Ã() (q )w q 2 φχ dx. R R β B(0,)\B(0, R )

17 6 J.M. ARRIETA, A. RODRIGUEZ-BERNAL AND J.D. ROSSI Taking the limit as 0 and evaluating at w = u we obtain (by [] or by the results of the previous section) F (u, 0)(φ)(χ) = χ φ dx + R 2 χφ dx w B(0,) B(0,) R 2 T q (q )u q 2 φχ dx. R β B(0,) This problem corresponds exactly with the linearized of (3.2) that is invertible by our choice R < R 0. Therefore, by the implicit function theorem, we get that there exists 0 such that for any < 0 there exists a unique solution w S of F (w, ) = 0 close to u, that is, a unique weak solution of (3.4), with lim w = u. Since we have proved that every extremal of (.4) goes to u as 0 and we have uniqueness of solutions of (3.4) in a neighborhood of u, then the extremals must be radial. Acknowledgements. The first and second authors have been partially supported by Grant BFM , Spain and the third author by EX066, CONICET and ANPCyT PICT No , Argentina. References [] J. M. Arrieta, A. Jimenez-Casas and A. Rodriguez-Bernal. Nonhomogeneous flux condition as limit of concentrated reactions. Preprint [2] T. Aubin. Équations différentielles non linéaires et le problème de Yamabe concernant la courbure scalaire. J. Math. Pures et Appl., 55 (976), [3] R.J. Biezuner. Best constants in Sobolev trace inequalities. Nonlinear Analysis, 54 (2003), [4] P. Cherrier, Problèmes de Neumann non linéaires sur les variétés Riemanniennes. J. Funct. Anal. Vol. 57 (984), [5] O. Druet and E. Hebey. The AB program in geometric analysis: sharp Sobolev inequalities and related problems. Mem. Amer. Math. Soc. 60 (76) (2002). [6] M. del Pino and C. Flores. Asymptotic behavior of best constants and extremals for trace embeddings in expanding domains. Comm. Partial Differential Equations, 26 (-2) (200), [7] J. F. Escobar. Sharp constant in a Sobolev trace inequality. Indiana Univ. Math. J., 37 (3) (988),

18 SOBOLEV TRACE CONSTANT 7 [8] J. Fernández Bonder, E. Lami Dozo and J.D. Rossi. Symmetry properties for the extremals of the Sobolev trace embedding. Ann. Inst. H. Poincaré. Anal. Non Linéaire, 2 (2004), no. 6, [9] J. Fernández Bonder and J.D. Rossi. Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains. Comm. Pure Appl. Anal., (3) (2002), [0] J. Fernández Bonder and J.D. Rossi. on the existence of extremals for the Sobolev trace embedding theorem with critical exponent. Bull. London Math. Soc., 37() (2005), [] O. Ladyzhenskaya and N. Uralseva. Linear and Quasilinear Elliptic Equations, Academic Press (968). [2] E. Lami Dozo and O. Torne, Symmetry and symmetry breaking for minimizers in the trace inequality. Comm. Contemp. Math., 7(6) (2005), [3] Y. Li and M. Zhu. Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries. Comm. Pure Appl. Math., 50 (997), [4] S. Martinez and J.D. Rossi. Isolation and simplicity for the first eigenvalue of the p-laplacian with a nonlinear boundary condition. Abst. Appl. Anal., 7 (5), (2002), [5] A. Rodríguez-Bernal and A. Vidal-Lopez. Extremal equilibria for parabolic nonlinear reaction diffusion equations. Proceedings of the Equadiff 2005, Bratislava, Eslovakia. [6] A. Rodríguez-Bernal and A. Vidal-Lopez. Extremal equilibria for nonlinear parabolic equations and applications. Universidad Complutense de Madrid. Preprint MA-UCM [7] M. W. Steklov, Sur les problèmes fondamentaux en physique mathématique, Ann. Sci. Ecole Norm. Sup., Vol. 9 (902), [8] A. Vidal-Lopez, Soluciones extremales para problemas parabolicos de evolucion no lineales y aplicaciones, Ph.D. thesis, Departamento de Matemática Aplicada, Universidad Complutense de Madrid, J.M. Arrieta and A. Rodríguez-Bernal Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Madrid España. address: arrieta@mat.ucm.es, arober@mat.ucm.es Web page: jarrieta arober J. D. Rossi Instituto de Matemáticas y Física Fundamental Consejo Superior de Investigaciones Científicas Serrano 23, Madrid, Spain, on leave from Departamento de Matemática, FCEyN UBA (428) Buenos Aires, Argentina. address: jrossi@dm.uba.ar Web page: jrossi

19 PREPUBLICACIONES DEL DEPARTAMENTO DE MATEMÁTICA APLICADA UNIVERSIDAD COMPLUTENSE DE MADRID MA-UCM ON THE FUNDAMENTAL SOLUTION OF A LINEARIZED UEHLING-UHLENBECK EQUATION. M.Escobedo, S. Mischler and J.J.L. Velázquez. 2. MATHEMATICAL ANALYSIS AND STABILITY OF A CHEMOTAXIS MODEL WITH LOGISTIC TERM. J.I. Tello 3. UNIQUENESS AND COLLAPSE OF SOLUTIONS FOR A MATHEMATICASL MODEL WITH NONLOCAL TERMS ARISING IN GLACIOLOGY. A. Muñoz and J.I. Tello 4. MATHEMATICAL ANALYSIS, CONTROLABILLITY AND NUMERICAL SIMULATION OF A SIMPLE MODEL OF TUMOUR GROWTH. J.I. Díaz and J.I. Tello 5. AN INTEGRO-DIFFERENTIAL EQUATION ARISING AS A LIMIT OF INDIVIDUAL CELL-BASED MODELS. M.Bodnar and J.J.L.Velázquez 6. ON THE PATH OF A QUASI-STATIC CRACK IN MODE III. G.E.Oleaga 7. DISSIPATIVE PARABOLIC EQUATIONS IN LOCALLY UNIFORM SPACES J.M. Arrieta, J.W. Cholewa, T. Dlotko and A. Rodríguez.-Bernal 8. ON THE NEWTON PARTIALLY FLAT MINIMAL RESISTANCE BODY TYPE PROBLEM, M.Comte and J.I.Díaz 9. SPECIAL FINITE TIME EXTINCTION IN NONLINEAR EVOLUTION SYSTEMS: DYNAMIC BOUNDARY CONDITIONS AND COULOMB FRICTION TYPE PROBLEMS, J.I.Díaz 0. ON THE HAIM BREZIS PIONEERING CONTRIBUTIONS ON THE LOCATION OF FREE BOUNDARIES, J.I.Díaz. COMPLETE AND ENERGY BLOW-UP IN PARABOLIC PROBLEMS WITH NNLINEAR BOUNDARY CONDITIONS, P.Quittner and A.Rodríguez-Bernal 2. DYNAMICS OF A REACTION DIFFUSION EQUATION WITH A DICONTINUOUS NONLINEARITY, J.M.Arrieta, A.Rodríguez-Bernal and J.Valero 3. BIFURCATION AND STABILITY OF EQUILIBRIA WITH ASYMPTOTICALLY LINEAR BOUNDARY CONDITIONS AT INFINITY, J.M.Arrieta, R.Pardo and A.Rodríguez-Bernal 4. SINGULAR PERTURBATION ANALYSIS OF camp SIGNALING IN DYCTIOSTELIUM DISCOIDEUM AGGREGATES, G.Litcanu and J.J.L.Velázquez 5. ON THE FINITE DIMENSION OF ATTRACTORS OF PARABOLIC PROBLEMS IN R^ N WITH GENERAL POTENTIALS, J.M. Arrieta, N.Moya and A.Rodríguez-Bernal. 6. MINIMAL PERIODS OF SEMILINEAR EVOLUTION EQUATIONS WITH LIPSCHITZ NONLINEARITY, J. C. Robinson and A.Vidal-López 7. ON THE ESHELBY-KOSTROV PROPERTY FOR THE WAVE EQUATION IN THE PLANE, M.A.Herrero. G.E.Olega and J.J.L.Velázquez 8. NUMERICAL STUDY OF A FRACTIONAL SINE-GORDON EQUATION, G.Alfimov, T.Pierantozzi and L.Vázquez. 9. ON GENERALIZED FRACTIONAL EVOLUTION-DIFFUSION EQUATIONS, A.A.Kilbas, T.Pierantozzi, J.J.Trujillo and L.Vázquez 20. GLOBAL STABILITY AND BOUNDS FOR COARSENING RATES WITHIN THE MEAN- FIELD THEORY FOR DOMAIN COARSENING, B. Niethamer and J.J.L.Velázquez

20 2. SELF-SIMILAR BEHAVIOR FOR NONCOMPACTLY SUPPORTED SOLUTIONS OF THE LSW MODEL, J.J.L.Velázquez 22. ON THE DYNAMICS OF THE CHARACTERISTIC CURVES FOR THE LSW MODEL, J.J.L.Velázquez 23. PULLBACK ATTRACTORS AND EXTREMAL COMPLETE TRAJECTORIES FOR NON- AUTONOMOUS REACTION-DIFFUSION PROBLEMS, J.C.Robinson, A.Rodríguez-Bernal and A.Vidal-López 24. NASH EQUILIBRIA IN NONCOOPERATIVE PREDATOR-PREY GAMES, A.M.Ramos and T.Roubicek 25. AN INTERPOLATION BETWEEN THE WAVE AND DIFFUSION EQUATION THROUGH THE FRACTIONAL EVOLUTION EQUATIONS DIRAC LIKE, T.Pierantozzi and L.Vazquez.

21 PREPUBLICACIONES DEL DEPARTAMENTO DE MATEMÁTICA APLICADA UNIVERSIDAD COMPLUTENSE DE MADRID MA-UCM ON THE BOUNDEDNESS OF SOLUTIONS OF REACTION DIFFUSION EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS, J.M. Arrieta 2. THE CLASSICAL THEORY OF UNIVALENT FUNCTIONS AND QUASISTATIC CRACK PROPAGATION, Gerardo E. Oleaga. 3. INVERTIBLE CONTRACTIONS AND ASYMPTOTICALLY STABLE ODE S THAT ARE NOT C^ -LINEARIZABLE, Hildebrando M. Rodrigues and Joan Solá-Morales. 4. CONVERGENCIA DE PROCESOS ALEATORIOS Y TEOREMAS L ÍMITE EN TEOR ÍA DE PROBABILIDADES, Ignacio Bosch Vivancos 5. A DYNAMICS APPROACH TO THE COMPUTATION OF EIGENVECTORS OF MATRICES, S.Jiménez and L.Vázquez 6. EXTREMAL EQUILIBRIA FOR REACTION DIFFUSION EQUATIONS IN BOUNDED DOMAINS AND APPLICATIONS, A.Rodríguez-Bernal and A.Vidal-López 7. EXISTENCE, UNIQUENESS AND ATTRACTIVITY PROPERTIES OF POSITIVE COMPLETE TRAJECTORIES FOR NON-AUTONOMOUS REACTION-DIFFUSION PROBLEMS, A.Rodríguez-Bernal and A.Vidal-López 8. DYNAMICS IN DUMBBELL DOMAINS I. CONTINUITY OF THE SET OF EQUILIBRIA, José M. Arrieta, Alexandre N. Carvalho and German Lozada-Cruz 9. INSENSITIZING CONTROLS FOR THE -D WAVE EQUATION, René Dager 0. THE BEST SOBOLEV TRACE CONSTANT AS LIMIT OF THE USUAL SOBOLEV CONSTANT FOR SMALL STRIPS NEAR THE BOUNDARY, J.M. Arrieta, A. Rodríguez- Bernal and J.Rossi