GINZBURG-LANDAU FUNCTIONALS AND RENORMALIZED ENERGY: A REVISED Γ-CONVERGENCE APPROACH

Size: px
Start display at page:

Download "GINZBURG-LANDAU FUNCTIONALS AND RENORMALIZED ENERGY: A REVISED Γ-CONVERGENCE APPROACH"

Transcription

1 GINZBURG-LANDAU FUNCTIONALS AND RENORMALIZED ENERGY: A REVISED Γ-CONVERGENCE APPROACH ROBERTO ALICANDRO AND MARCELLO PONSIGLIONE Abstract. We give short and self-contained proofs of Γ-convergence results for Ginzburg- Landau energy functionals in dimension two, in the logarithmic energetic regime. In particular, we derive the renormalized energy by Γ-convergence. Keywords: Ginzburg-Landau theory, Topological singularities, Calculus of Variations. 000 Mathematics Subject Classification: 35Q56, 58K45, 49J45, 35J0.. Introduction The analysis of Ginzburg-Landau functionals as the length-scale parameter tends to zero is a beautiful piece of mathematical analysis. Different ideas and techniques merged together along the last twenty years, to give a clear picture of the relevant phenomena, as concentration of energy and formation of topological singularities. To make a long story very short, the analysis started with the study of the asymptotic behaviour of the minimizers of Ginzburg-Landau functionals in dimension two, with prescribed boundary conditions. Let Ω R be bounded and with Lipschitz boundary. The Ginzburg-Landau functionals GL : H (Ω; R ) [0, + ), are defined as ( (.) GL (u) := u + ) W (u), where W C 0 (R ) is such that W (x) 0, W {0} = S and lim inf x Ω W (x) W (x) > 0, lim inf ( x ) x x > 0. In [] are collected the main results for the asymptotic behavior of the minimizers u of GL with a prescribed boundary datum g : Ω S with degree d. For small, the energy GL (u ) blows up as d log, and d vortex-like singularities appear, around which u looks like (a fixed rotation of) x/ x. Subtracting the leading term d log from the energy, a finite quantity remains in the limit, called renormalized energy, depending on the position of the singularities. After these results, much work has been devoted to understand the behavior of sequences of non minimizers, with prescribed energetic regime, in the spirit of Γ-convergence. The main issues are GL clearly the (zero order) Γ-convergence of log and the (first order) Γ-convergence of GL d log to the renormalized energy. The picture is nowadays well understood. For the zero order Γ- convergence [8] and [6] provide sharp lower bounds, while in [6] and [7] a Γ-convergence result is proved in W, (Ω; R ), and compactness of the singularities is expressed in terms of compactness properties of the Jacobians Ju of u in the dual of Hölder functions; in [] the Γ-convergence result is obtained (in any dimension and codimension) with respect to the convergence of the Jacobians. To our knowledge a self contained proof of the first order Γ-convergence result is still missing. The aim of this paper is to revisit these Γ-convergence results, giving short, efficient and selfcontained proofs. Self-contained has to be understood in a very weak sense: we use many ideas from [], [8], [6], [7] and (for the first order Γ-limit) the analysis for minimizers developed in []. Our approach is the following: we consider the ball construction as done in [8]: it consists in an efficient way of selecting balls where the energy concentrates. Then we plug a Dirac mass in

2 R. ALICANDRO AND M. PONSIGLIONE each ball, obtaining a sequence of measures µ that approximates Ju, carrying all the topological information and bringing compactness and Γ-liminf inequality in the zero order Γ-convergence result. The Γ-limsup inequality is obtained by a standard construction. To prove the first order Γ-convergence result we show that, if GL (u )/ log is bounded, then around each singularity x i we have GL (u ) πz i log C, where z i Z is the degree of the singularity. Moreover, if u are optimal in energy, then z i = and around each singularity u looks like (a rotation of) x x. These preliminary results allow to use the analysis done in [] to derive the renormalized energy. Let us conclude recalling that the case treated in this paper has been the building block for a series of important generalizations, as for the case of external magnetic field [9], the three dimensional case [] and the study of the evolution of vortices [0]. The final goal of this paper is to rewrite this building block within the solid formalism of Γ-convergence, with efficient and self-contained short proofs.. Preliminary results In this section we introduce the notion of Jacobian, current and degree. Given u H (Ω; R ), the Jacobian Ju of u is the L function defined by Ju := det u. Let us denote by C 0, (Ω) the space of Lipschitz continuous functions on Ω endowed with the norm ϕ C 0, := sup f(x) + x Ω sup x, y Ω, x y ϕ(x) ϕ(y), x y and by Cc 0, (Ω) its subspace of functions with compact support. The norm in the dual of Cc 0, (Ω) will be denoted by and referred to as norm, while denotes the convergence in the norm. Finally, the norm in the dual of C 0, (Ω) will be denoted by (Ω) and by (Ω) the corresponding convergence. For every u H (Ω; R ), we can consider Ju as an element of the dual of C 0, (Ω) by setting < Ju, ϕ >:= Ju ϕ dx for every ϕ C 0, (Ω). Ω Notice that Ju can be written in a divergence form as Ju = div (u (u ) x, u (u ) x ), i.e., for every ϕ Cc 0, (Ω), (.) < Ju, ϕ >= u (u ) x ϕ x u (u ) x ϕ x dx. Equivalently, we have Ju = curl (u u ) and Ju = curl j(u), where Ω j(u) := (u u + u u ) is the so called current. Notice that if u L (Ω; R ), then Ju is in the dual of H (Ω). Let A Ω open with Lipschitz boundary. Then we have (.) Ju = curl j(u) := j(u) τ, A A A where τ is the tangent field to A, and the last integral is meant in the sense of H. Let h H ( A; R ) with h α > 0. The degree of h is defined as follows (.3) deg(h, A) := j(h/ h ) τ. π In [4], [5] it is proved that the definition is well posed, that deg(h, A) Z and, whenever u H (A; R ), u β > 0 in A, deg(u, A) = 0, where in the notation of degree we identify u with its trace. Finally, deg(u, A) is stable with respect to the strong convergence in H (A; R ). Notice that u can be written in polar coordinates as u(x) = ρ(x)e iθ(x) on A with ρ α, where θ is the so called lifting of u. By [3, Theorem ] (see also [3, Remark 3]), if A is simply connected A

3 GINZBURG-LANDAU FUNCTIONALS AND RENORMALIZED ENERGY 3 and deg(u, A) = 0, then the lifting can be selected in H ( A) with the map u θ continuous (but the image of a bounded subset of H ( A; S ) is not necessarily bounded in H ( A)). If the degree d is not zero, then the lifting can be locally selected in H ( A) with a jump of order πd. Now we introduce some notion of modified Jacobian we will use in our Γ-convergence results. The main observation is that, for every v := (v, v ), w := (w, w ) belonging to H (Ω; R ) we have (.4) Jv Jw = ( J(v w, v + w ) J(v w, v + w ) ). By (.) and (.4) we immediately deduce the following lemma. Lemma.. Let v n and w n be two sequences in H (Ω; R ) such that Then Jv n Jw n 0. v n w n ( v n + w n ) 0. Given 0 < τ < and u H (Ω; R ), set (.5) u τ := T τ ( u ) u u, J τ u := Ju τ, where T τ (ρ) = min{ ρ τ, }. By Lemma. we easily deduce the following proposition. Proposition.. Let u be a sequence in H (Ω; R ) such that GL (u ) C log, and let 0 < s < be such that as 0. Then s log sup Ju J s u 0 as 0. s s 3. Ball construction In this section we revisit the celebrated ball construction, a useful machinery for providing lower bounds, following the approach by Sandier [8]. Let B = {B r (x ),..., B rm (x m )} be a finite family of disjoint balls in R, and set Rad(B) := ri. The ball construction consists in letting the balls alternatively expand and merge each other. The expansion phase consists in letting all the balls expand in such a way that at each (artificial) time the ratio θ(t) := r i (t)/r i is independent of i; we will parametrize the time enforcing θ(t) = + t. The expansion phase stops at the first time T when two balls B ri(t)(x i ), B rj(t)(x j ) touch each other. Then the merging phase begins. It consists in collecting the balls B ri(t )(x i ) in subclasses and merging all the balls of a subclass in a larger ball B Rj (y j ) with the following properties. i) R j is not larger than the sum of all the radii of the balls B ri(t )(x i ) contained in B Rj (y j ). ii) The balls B Rj (y j ) of the new family are disjoint. After the merging, we define in each ball B Rj (y j ) a seed size s j by R j /s j = θ(t ) = + T (we set s j = r j for t = 0). Then another expansion phase begins, during which we keep the seed sizes constant, and we now enforce θ(t) := R j (t)/s j = + t, where t T. We proceed so forth alternating merging to expansion phases, until a last phase where only one ball expands. Notice that by construction, in particular by property i), s j does not increase during the merging. In particular, we always have (3.) Rj (t) = ( + t)s j ( + t)rad(b). Now assume that to each ball B ri (x i ) of the original family B corresponds some integer multiplicity z i Z, and set µ := i z iδ xi. Let F (B, µ, U) be defined as follows: if A r,r (x) := B R (x)\b r (x) is an annulus that does not intersect any B ri (x i ), we set ( ) R F (B, µ, A r,r (x)) := π µ(b r (x)) log. r

4 4 R. ALICANDRO AND M. PONSIGLIONE Then, for every open set U R we set F (B, µ, U) := sup i F (A i ), where the sup is over all finite families of disjoint annuli A i U that do not intersect any B ri (x i ). Remark 3.. The definition of F is justified by the following observation. Let Ω = Ω \ B B B. Given u H ( Ω; S ), let µ := B C deg(u, B)δ x, where C denotes the family of balls in B that are contained in Ω, and x is the center of B. Then, by Jensen inequality we easily deduce that F (B, µ, U) u, for every open set U Ω (see for instance [8]). Set B(t) the family of balls at time t (with the convention B(t) = B(t ) if t is a merging time). By our definitions it easily follows (see [8] for more details) that for any B B(t) U Ω (3.) F (B, µ, B) π µ(b) log( + t). We introduce the modified measure µ as follows: let C() be the family of balls in B() that are contained in Ω. We set (3.3) µ := µ(b r (x))δ x. B r(x) C() Theorem 3.. Let B n = {B ri,n (x i,n )} be a sequence of finite families of disjoint balls in R with Rad(B n ) 0, and let µ n := i z i,nδ xi,n, z i,n Z, be a given multiplicity measure. Assume (3.4) F (B n, µ n, Ω) C log Rad(B n ). Then the following holds. ) Up to a subsequence µ n µ for some µ = N ) Assume µ n µ = N i= z iδ xi with z i Z, x i Ω. i= z iδ xi. For every i and every σ dist(x i, Ω j i x j ) σ lim inf F (B n, µ n, B σ (x i )) π z i log n Rad(B n ) 0. Proof. We let the families B n grow and merge as described above, and set B n (t) the corresponding family of balls at time t. Set C n (t) := {B B n (t), B Ω}, D n (t) := {B B n (t), B Ω }. Set moreover t n := Rad(Bn ), ν n := µ n (B r (x))δ x. B r(x) C n(t n) By the energy bound (3.4) and by (3.) we have µ n (Ω) C log Rad(B n ), ν n (Ω) C, and hence up to a subsequence ν n µ for some µ = N i= z iδ xi. To conclude the proof of ) it is enough to prove that ν n µ n 0. By construction, (ν n µ n )(B) = 0 for every B C n (t n ). We conclude that for every ϕ n Cc 0, (Ω) with ϕ n C 0, we have c (3.5) < ν n µ n, ϕ n >= ϕ n d(ν n µ n ) + ϕ n d(ν n µ n ) B C n(t n) (max B B B n(t n) B C n(t n) B B D n(t n) ϕ n min B ϕ n) ( ν n + µ n )(B) + B D n(t n) B max ϕ n ( ν n + µ n )(B) B diam(b)( ν n + µ n )(B) C Rad(B n ) log Rad(B n ) 0 as n, where in the last inequality we have used (see (3.)) Rad(B n (t n )) Rad(B n ). We pass to the proof of ). Let 0 < δ < σ be fixed. For n large enough we have that all the balls B B n (t n ) with µ n (B) 0 either have a distance from x i smaller then δ, or larger than σ δ. We call the family of balls enjoying the first property G n, and H n the others. Notice that, for n

5 GINZBURG-LANDAU FUNCTIONALS AND RENORMALIZED ENERGY 5 large enough, B G n µ n (B) = z i. Let t n = σ δ Rad(B. Let I n) n be the family of balls B B n ( t n ) which are contained in B σ (x i ). By (3.) we have diam(b) σ δ for every B B n ( t n ), and hence the balls in I n contain all the balls in G n (and none in H n ). Recalling (3.) we deduce that F (B n, µ n, B σ (x i )) B I n π µ n (B) log( + t n ) π z i log( + t n ) = π z i log σ δ Rad(B n ). We conclude by letting n + and δ Γ-convergence of GL In this section we prove the zero order Γ-convergence result of GL that collects results proved in [6], [7], []. We will use the standard notation GL (u, A) to denote the Ginzburg-Landau functional localized on the open set A, i.e., defined as in (.) with Ω replaced by A. Theorem 4.. The following Γ-convergence result holds. i) (Compactness) Let (u ) H (Ω; R ) be such that GL (u ) C log. Then, up to a subsequence, J(u ) µ, where µ := π N i= z iδ xi for some x i Ω, z i Z. ii) (Γ-liminf inequality) Let (u ) H (Ω; R ) be such that J(u ) µ := π N i= z iδ xi. Then, there exists C R such that, for every i and every σ dist(x i, Ω j i x j ) we have (4.) lim inf GL (u, B σ (x i )) π z i log σ C. In particular, there exists a constant C such that (4.) lim inf GL (u ) log µ (Ω) C. iii) (Γ-limsup inequality) For every µ := π N i= z iδ xi, there exists (u ) H (Ω; R ) such that J(u ) µ, log GL (u ) µ (Ω), and if z i =, GL (u ) µ (Ω) log C for some C R. Proof. By standard density arguments in Γ-convergence we may assume that u are smooth. Following the notations in [8], we set Ω,τ := { u > τ}, γ,τ := Ω,τ \ Ω, Θ (τ) := u u, n (τ) := u. γ,τ By Coarea Formula we have ( GL (u ) 0 n (τ) + Ω,τ ) γ,τ u dτ 0 τ dθ (τ), where Θ (τ) is the distributional derivative of the decreasing function Θ (τ) and the inequality is due to the possible presence of regions { u = 0} with positive measure. Set τ := /3, and K,τ := Ω \ Ω,τ. Then, by the energy bounds, for every τ τ we have K,τ C 4 3 log Ω, so that, using also that Ω is Lipschitz we have (4.3) H ( K,τ ) CH (γ,τ ). Notice that, by definition of Hausdorff measure, being K,τ compact, it is always contained in a finite union of balls B ri (y i ) such that i r i H ( K,τ ). Moreover, after a merging procedure, we can always assume that such balls are disjoint. As a consequence, either K,τ or Ω \ K,τ is contained in the union of such balls. In the latter case, since Ω \ K,τ Ω we have i r i c, and we replace these balls by one single ball containing Ω. In both cases, we have a family of balls B,τ whose union contains K,τ, such that (4.4) Rad(B,τ ) ch ( K,τ ) CH (γ,τ ). Let K,τ be their union and Ω,τ := Ω \ K,τ. Since K,τ is monotone in τ (with respect to inclusion), we can always assume that Rad(B,τ ) is increasing with respect to τ.

6 6 R. ALICANDRO AND M. PONSIGLIONE By Hölder inequality and (4.4), for τ τ we have (4.5) Rad(B,τ ) CH (γ,τ ) Cn (τ) γ,τ u. By (4.5), Young inequality and integration by parts of τ dθ (τ) we have (4.6) τ 0 GL (u ) n (τ) + H (γ,τ ) 0 C dτ τ dθ n (τ) (τ) 0 H (γ,τ ) + τθ (τ) dτ 0 C τ Rad(B,τ ) + τθ (τ) dτ + Rad(B,τ ) + τθ (τ) dτ C τ C where τ = 3. By (4.6) and the energy bound, recalling also that τ Rad(B,τ ) is increasing, it follows that (4.7) Rad(B,s ) Rad(B, τ ) C 3 log for all s τ. Let C,s be the family of balls in B,s that are contained in Ω, and set µ,s := B C,s deg(u, B)δ x, where x is the center of B. Moreover, let µ,s be the corresponding modified measure defined according with (3.3), i.e., letting C,s () be the family of balls in B,s () that are contained in Ω, µ,s = µ (B)δ x, B C,s() where x is the center of B. By (4.6) and (4.7) we have θ ( τ ) C log C log Rad(B, τ ). Since θ (s) F (B,s, µ,s, Ω) (see Remark 3.), by the energy bound and Theorem 3. we have (up to a subsequence) (4.8) π µ, τ µ := π N z i δ xi. By definition of µ,s (see also (.) and (.5)) we have that (J s u π µ,s )(B) = 0 for every B C,s (). Therefore, By Proposition (.) and (4.7) we have (4.9) lim 0 7 i= lim sup J(u ) π µ,s = lim s τ sup s τ B B,s() ϕ C 0, c sup J s (u ) π µ,s = 7 s τ J s u (B) osc B (ϕ) lim 0 C 7 log Rad(B, τ ()) = 0. By (4.8) and (4.9) we deduce in particular the compactness property i). Now we prove the Γ-liminf inequality, starting with the global version (4.). By Theorem 3. and (4.9) we have (4.0) lim inf From (4.6) we deduce GL (u ) τ 7 inf Θ (s) µ (Ω) log Rad(B,s ) C. 7 s τ Rad(B,τ ) τ µ (Ω) log Rad(B,τ ) dτ C C

7 GINZBURG-LANDAU FUNCTIONALS AND RENORMALIZED ENERGY 7 Notice that the right-hand side is minimized for Therefore, for small enough, (4.) GL (u ) τ µ (Ω) log µ (Ω) Rad(B,τ ) = C µ (Ω)τ. 7 τ µ (Ω)τ τ µ (Ω) log C µ (Ω)τ 7 dτ C log C µ (Ω)τ dτ C µ (Ω) log C. The proof of (4.) is identical, replacing Ω by B σ (x i ) and (4.0) by lim inf sup Θ (s) π z i (log 7 s τ σ Rad(B,s ) ) C. Let us sketch the proof of the Γ-limsup inequality. By a standard density argument we can assume z i = ±. Let u,i (r, θ) := e ±iθ g(r), where (θ, r) are polar coordinate centered at x i and g(s) := min{ s, }. Then a recovery sequence is given by u = ΠN i= u i, where u i are identified with complex functions. The straightforward computations are left to the reader. 5. First order Γ-convergence to the renormalized energy In this section we prove the first order Γ-convergence of GL to the renormalized energy, introduced in []. We begin by recalling the main definitions and results of [] that we need. Let µ := N i= z iδx i, with z i =, x i Ω. Let moreover Φ 0 be the solution of { Φ0 = πµ in Ω, Φ = 0 on Ω, and let R 0 (x) = Φ 0 d i log x x i. The renormalized energy is defined as follows (5.) W(µ) := π i j z i z j log x i x j π i z i R 0 (x i ). Let now σ > 0 be such that B σ (x i ) are disjoint and contained in Ω and set Ω σ := Ω \ i B σ (x i ). Consider the following minimization problems { } (5.) m(σ, µ) := min u, deg(u, B σ (x i )) = z i, u H (Ω σ;s ) Ω { σ m(σ, µ) := min u, u(z) = α } i (5.3) u H (Ω σ;s ), α i = Ω σ σ (z x i) zi on B σ (x i ), zi { γ(, σ) := min GL (u, B σ ), u(x) B σ = x } (5.4). u H (B σ;r ) x Theorem 5. (Bethuel, Brezis, Hélein []). We have lim m(σ, µ) + π µ (Ω) log σ = lim m(σ, µ) + π µ (Ω) log σ = W(µ). σ 0 σ 0 Moreover, there exists γ R such that lim γ(, σ) + π log 0 σ = γ. Remark 5.. Consider the case Ω = B R, µ = δ 0. by Jensen inequality we have that the minimizers of (5.) are given by the one parameter family { (5.5) K := α z }, α C, α =. z In particular, by Theorem 5., for Ω = B R we have W(δ 0 ) = π log R. Theorem 5.3. The following Γ-convergence result holds.

8 8 R. ALICANDRO AND M. PONSIGLIONE i) (Compactness) Let M N and (u ) H (Ω; R ) be such that GL (u ) Mπ log C. Then, up to a subsequence, Ju µ = N i= z iδ xi, with z i Z \ {0}, x i Ω, Ñ := zi M. Moreover, if Ñ = M, then N = Ñ = M, i.e., z i = for every i. ii) (Γ-liminf inequality) Let u be such that Ju µ = M i= z iδ xi with z i =, x i Ω. Then, lim inf GL (u ) Mπ log W(µ) + Mγ. iii) (Γ-limsup inequality) Given µ = M i= z iδ xi, z i =, x i Ω, there exists u with Ju µ such that GL (u ) Mπ log W(µ) + Mγ. Proof. Proof of i). The fact that, up to a subsequence Ju µ = N i= z iδ xi, with Ñ := z i M, is a direct consequence of the zero order Γ-convergence result stated in Theorem 4.. Assume now Ñ = M, and let us prove that z i =. Let 0 < σ < σ be such that that B σ (x i ) are disjoint and contained in Ω. Then, by (4.) in Theorem 4., for small enough we have (5.6) N N GL (u ) GL (u, B σ (x i )) + GL (u, A i ) π z i log σ + GL (u, A i ) C, i= where A i := B σ (x i ) \ B σ (x i ). By the energy bound, we deduce that GL (u, A) C, and hence (up to a subsequence) u u i in H (A i ; R ), for some u i = e iθi(x) H (A i ; S ). Moreover, it is easy to see that deg(u i, B σ (x i )) = z i, for instance arguing as follows: by the energy bound and standard Fubini s arguments, for almost every σ < σ < σ we have that (up to a subsequence) the trace of u on B σ (x i ) is bounded in H ( B σ (x i ); R ) and hence weakly converge to the trace of u. The assertion follows by the very definition of degree (.3). For every i we have (5.7) A i u i = By (5.7) and (5.6) we conclude that, for small enough, GL (u ) N i= i= A i θ i π z i (log σ log σ ). π z i log σ + π z i (log σ log σ ) C = πm log + π N (zi z i ) log σ C. σ Letting σ 0, the energy bound yields z i. Proof of ii). For every r > 0, by (4.) we can assume that GL (u, Ω r ) C, so that (up to a subsequence) u u in Hloc (Ω \ x i; R ). Let σ > 0 be such that B σ (x i ) are disjoint and contained in Ω. Let t σ, and consider the minimization problem (5.) in B t \ B t for µ = δ 0. Set d t (w, K) := min{ w v H (B t\b t ;R ) : v K}, where K is the family of minimizers given by (5.5). It is easy to prove that for any given δ > 0 there exists c > 0 (independent of t) such that, if d t (u ( + x i ), K) δ, then lim inf u log + c. B t(x i)\b t (x i) Indeed, arguing by contradiction, if there exists a subsequence u such that log u lim u = log, B t(x i)\b t (x i) B t(x i)\b t (x i) then we would conclude that u u strongly in H (B t (x i )\B t (x i); R ), hence d t (u( +x i ), K) δ. This is in contradiction with the definition of K (see Remark 5.), noticing that deg(u i, B t (x i )) = z i = ± and that u( + x i ) has minimal energy log. Let m N be such that mc W(µ) + M(γ log σ C), where C is the constant in (4.). For l =,..., m set C l (x i ) := B l σ(x i ) \ B l σ(x i ). We consider now two cases. i=

9 GINZBURG-LANDAU FUNCTIONALS AND RENORMALIZED ENERGY 9 First case: for small enough and for every fixed l m, there exists at least one i such that d σ(u l ( + x i ), K) δ. Then by (4.) we conclude (for small enough) (5.8) GL (u ) M(log σ m log + C) + m M l= i= u C l (x i) M(log σ m log log + C) + m ( (M log + c ) M(log σ log + C) + W(µ) + M(γ log σ C) = M( log + γ) + W(µ). Second case: Up to a subsequence, there exists l m such that for every i we have d σ (u ( + x i ), K) δ, where σ := lσ. By standard Fubini s arguments there exists /3 σ σ 3/4 σ such that the Ginzburg-Landau energy of u on B σ (x i ) is uniformly bounded. We can easily modify u with an infinitesimal amount of energy to enforce u = on B σ (x i ). Let us denote by θ the lifting of x/ x. By [3, Theorem ] (see also [3, Remark 3]) there exists r(δ) 0 as δ 0 such that u = u e iθ (up to additive constant in the phase) with θ θ H ( B σ(x i)) r(δ). Let T be the harmonic solution on B σ (x i )\B σ (x i ) with boundary conditions θ and θ on B σ (x i ) and B σ (x i ), respectively. We extend u on B σ (x i ) \ B σ (x i ) setting ũ := e it on B σ (x i ) \ B σ (x i ). Hence we have GL (u, B σ (x i )) GL (ũ, B σ (x i )) + r(, δ) γ(, σ) + r(, δ), where lim δ 0 lim 0 r(, δ) = 0. By Theorem 5. we conclude that GL (u ) = GL (u, Ω σ ) + M GL (u, B σ (x i )) W(µ) Mπ log σ + M(γ π log ) + r(, δ) = M( log + γ)) + W(µ) + r(, δ). σ The proof follows by the arbitrariness of δ. Proof of iii). Let u,σ be the function that agrees with a minimizer of (5.3) in Ω σ, and with α i w,σ (x) in each B σ (x i ), where w,σ is a minimizer of problem (5.4), and α i is a unit vector suitable chosen to match the boundary conditions on B σ (x i ). By Theorem 5. there exists a suitable σ such that u := u,σ is optimal in energy. 6. Boundary conditions In this section we deal with prescribed boundary conditions g on Ω, and prove in this context the first order Γ-convergence of GL to the renormalized energy (the zero order Γ-convergence result can be deduced as a particular case). The renormalized energy W g is defined according with (5.), but now the function φ solves the following boundary value problem. { Φ = µ in Ω; Φ ν = g g τ on Ω. Then, Theorem 5. holds true with W replaced by W g, and with m(σ, µ), m(σ, µ) replaced by m g (σ, µ), m g (σ, µ) defined in the obvious way. Given g H ( Ω; R ) we denote by H g (Ω; R ) the subspace of functions in H (Ω; R ) with trace g. Theorem 6.. The following Γ-convergence result holds. i) (Compactness) Let M N, (u ) H g (Ω; R ) be such that GL (u ) Mπ log C. (Ω) Up to a subsequence, Ju µ = N i= z iδ xi, where x i Ω, z i Z \ {0}, z i = deg(g, Ω), Ñ := z i M. Moreover, if Ñ = M, then N = Ñ = M, i.e., z i = for every i, and all x i s belong to Ω. In particular, if M = deg(g, Ω), then z i = sign(deg(g, Ω)). i=

10 0 R. ALICANDRO AND M. PONSIGLIONE ii) (Γ-liminf inequality) Let u Hg (Ω; R ) be such that Ju µ = M i= z iδ xi with z i =, x i Ω. Then, lim inf GL (u ) Mπ log W g (µ) + Mγ. iii) (Γ-limsup inequality) Given µ = M i= z iδ xi with z i =, x i Ω, there exists u Hg (Ω; R ) with Ju µ such that GL (u ) Mπ log W g (µ) + Mγ. Proof. The proof of ii) and iii) is exactly as the proof of the analogous statements in Theorem 5.3, so we give only the proof of i). We can assume that g is the trace of a function, still denoted by g H (U; S ), where U is a neighborhood of Ω. Let Ω := Ω U, and let ũ be the extension of u on Ω defined by ũ = u in Ω and ũ = g on U \ Ω. By construction we have (6.) GL (ũ, Ω) Mπ log C for some C 0. There exists a costant c > 0 such that, given ϕ C 0, (Ω), there exists an extension ϕ Cc 0, ( Ω) of ϕ with ϕ C 0, c ( Ω) c ϕ C 0, (Ω). By Theorem 4. (using also that Jg = 0), there exists µ = N i= z iδ xi with x i Ω, Ñ := z i M, such that Ju µ (Ω) = Moreover, by (.) and (.3) we have sup < Ju µ, ϕ > ϕ C 0, (Ω) deg(g, Ω) = Ju (Ω) sup ϕ 0, C c ( Ω) c N z i. i= < Jũ µ, ϕ > 0. If now Ñ = M, we deduce z i = for every i by statement i) of Theorem 5.3. Finally, the fact that x i belong to Ω follows as in the proof of ii) of Theorem 5.3. More precisely, by (6.) we have that (up to a subsequence) ũ u in Hloc ( Ω \ x i ). Recall that for any t such that B t (x i ) Ω, given δ > 0 there exists c > 0 (independent of t) such that, if the distance of u from K in H (B t (x i ) \ B t (x i)) is larger than δ, then u log + c. B t(x i)\b t (x i) It follows that for any δ the distance of u from K is less then δ for infinitely many annuli B n(x i )\ B (n+)(x i ). This is possible only if x i belongs to Ω. Finally, we consider the case of varying boundary conditions g. We need the following Lemma. Lemma 6.. Let g, h H ( Ω; R ) be such that g h 0 in H ( Ω; R ), and such that g and g h converge to 0 uniformly as 0. Finally, let v Hg (Ω; R ) with v C, GL (v ) C log. Then, there exists w Hh (Ω; R ) such that GL (v ) GL (w ) 0 as 0. Proof. By [3, Theorem ] (see also [3, Remark 3]) we can always write v = v e iθ(x) and h = h e i(θ(x)+t) with t 0 in H ( Ω). Let τ be the harmonic extension of t in Ω, and let ρ be the harmonic extension of h g in Ω. Notice that ρ e iτ in H (Ω) and (ρ e iτ )/ 0 uniformly. The desired function w is given by w (x) := ρ (x)e i(τ(x)) v (x). Theorem 6.3. Let g H ( Ω; S ), g H ( Ω; R ) be such that i) g g in H ( Ω; R ); ii) g g 0 uniformly as 0.

11 GINZBURG-LANDAU FUNCTIONALS AND RENORMALIZED ENERGY Then, Theorem 6. still holds true with u H g (Ω; R ) replaced by u H g (Ω; R ). Proof. Any u Hg (Ω; R ) bounded in energy can be extended in a neighborhood Ω of Ω with a bounded amount of energy, so that the proof of compactness follows exactly as in the case of a fixed boundary datum. A possible extension is constructed as follows: by [3, Theorem ] (see also [3, Remark 3]) we have g = ge it with t 0 in H ( Ω). Then, we define ũ on Ω \ Ω as ũ = gρ e iτ, where g is an extension of g, ρ is the harmonic extension of g with boundary datum on Ω, and τ is the harmonic extension of t with boundary datum 0 on Ω. Finally, by Lemma 6. for any sequence u bounded in energy and in L one can easily switch the boundary conditions from g to g and viceversa with vanishing perturbations in the energy. This is enough to deduce the Γ-liminf and Γ-limsup inequality from Theorem 6.. Acknowledgments We wish to tank Etienne Sandier for some discussions and suggestions that made more elegant the proof of Theorem 5.3, and Adriano Pisante for many clarifying discussions on the theory of degree and lifting map. References [] Alberti G., Baldo S., Orlandi G.: Variational convergence for functionals of Ginzburg-Landau type. Indiana Univ. Math. J. 54 (005), no. 5, [] Bethuel F., Brezis H., Hélein F.: Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and their Applications. vol. 3. Birkhäuser Boston, Boston, 994. [3] Bourgain J., Brezis H., Mironescu P.: Lifting in Sobolev spaces. J. Anal. Math. 80 (000), [4] Boutet de Monvel-Berthier A., Georgescu V., Purice R.: A boundary value problem related to the Ginzburg- Landau model. Comm. Math. Phys. 4 (99), 3. [5] Brezis, H., Nirenberg, L.: Degree theory and BMO: Part i: compact manifolds without boundaries. Selecta Math. (N.S.) (995), no., [6] Jerrard R.L.: Lower bounds for generalized Ginzburg-Landau functionals. SIAM J. Math. Anal. 30 (999), no. 4, [7] Jerrard R. L., Soner H. M.: The Jacobian and the Ginzburg-Landau energy. Calc. Var. Partial Differential Equations 4 (00), no., 5 9. [8] Sandier, E.: Lower bounds for the energy of unit vector fields and applications. J. Funct. Anal. 5 (998), no., [9] Sandier E., Serfaty S.: Vortices in the Magnetic Ginzburg-Landau Model. Progress in Nonlinear Differential Equations and their Applications, 70. Birkhuser Boston, Inc., Boston, MA, 007. [0] Sandier E., Serfaty S.: Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Comm. Pure Appl. Math 57 (004), no, (R. Alicandro) DAEIMI, Università di Cassino via Di Biasio 43, Cassino (FR), Italy address, R. Alicandro: alicandr@unicas.it (Marcello Ponsiglione) Dipartimento di Matematica G. Castelnuovo, Sapienza Universitá di Roma, Piazzale A. Moro, 0085 Roma, Italy address, M. Ponsiglione: ponsigli@mat.uniroma.it

A Ginzburg-Landau approach to dislocations. Marcello Ponsiglione Sapienza Università di Roma

A Ginzburg-Landau approach to dislocations. Marcello Ponsiglione Sapienza Università di Roma Marcello Ponsiglione Sapienza Università di Roma Description of a dislocation line A deformed crystal C can be described by The displacement function u : C R 3. The strain function β = u. A dislocation

More information

Γ-CONVERGENCE OF THE GINZBURG-LANDAU ENERGY

Γ-CONVERGENCE OF THE GINZBURG-LANDAU ENERGY Γ-CONVERGENCE OF THE GINZBURG-LANDAU ENERGY IAN TICE. Introduction Difficulties with harmonic maps Let us begin by recalling Dirichlet s principle. Let n, m be integers, Ω R n be an open, bounded set with

More information

DEGREE AND SOBOLEV SPACES. Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg. Introduction

DEGREE AND SOBOLEV SPACES. Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 13, 1999, 181 190 DEGREE AND SOBOLEV SPACES Haïm Brezis Yanyan Li Petru Mironescu Louis Nirenberg Dedicated to Jürgen

More information

ON MAPS WHOSE DISTRIBUTIONAL JACOBIAN IS A MEASURE

ON MAPS WHOSE DISTRIBUTIONAL JACOBIAN IS A MEASURE [version: December 15, 2007] Real Analysis Exchange Summer Symposium 2007, pp. 153-162 Giovanni Alberti, Dipartimento di Matematica, Università di Pisa, largo Pontecorvo 5, 56127 Pisa, Italy. Email address:

More information

METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES IN TWO DIMENSIONS: A Γ-CONVERGENCE APPROACH

METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES IN TWO DIMENSIONS: A Γ-CONVERGENCE APPROACH METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES IN TWO DIMENSIONS: A Γ-CONVERGENCE APPROACH R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE Abstract. This paper aims at building

More information

arxiv: v1 [math.ap] 9 Oct 2017

arxiv: v1 [math.ap] 9 Oct 2017 A refined estimate for the topological degree arxiv:70.02994v [math.ap] 9 Oct 207 Hoai-Minh Nguyen October 0, 207 Abstract We sharpen an estimate of [4] for the topological degree of continuous maps from

More information

Interaction energy between vortices of vector fields on Riemannian surfaces

Interaction energy between vortices of vector fields on Riemannian surfaces Interaction energy between vortices of vector fields on Riemannian surfaces Radu Ignat 1 Robert L. Jerrard 2 1 Université Paul Sabatier, Toulouse 2 University of Toronto May 1 2017. Ignat and Jerrard (To(ulouse,ronto)

More information

Renormalized Energy with Vortices Pinning Effect

Renormalized Energy with Vortices Pinning Effect Renormalized Energy with Vortices Pinning Effect Shijin Ding Department of Mathematics South China Normal University Guangzhou, Guangdong 5063, China Abstract. This paper is a successor of the previous

More information

SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction

SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.

More information

ESTIMATES OF LOWER CRITICAL MAGNETIC FIELD AND VORTEX PINNING BY INHOMO- GENEITIES IN TYPE II SUPERCONDUCTORS

ESTIMATES OF LOWER CRITICAL MAGNETIC FIELD AND VORTEX PINNING BY INHOMO- GENEITIES IN TYPE II SUPERCONDUCTORS Chin. Ann. Math. 5B:4(004,493 506. ESTIMATES OF LOWER CRITICAL MAGNETIC FIELD AND VORTEX PINNING BY INHOMO- GENEITIES IN TYPE II SUPERCONDUCTORS K. I. KIM LIU Zuhan Abstract The effect of an applied magnetic

More information

Improved estimates for the Ginzburg-Landau equation: the elliptic case

Improved estimates for the Ginzburg-Landau equation: the elliptic case Improved estimates for the Ginzburg-Landau equation: the elliptic case F. Bethuel, G. Orlandi and D. Smets May 9, 2007 Abstract We derive estimates for various quantities which are of interest in the analysis

More information

GRADIENT THEORY FOR PLASTICITY VIA HOMOGENIZATION OF DISCRETE DISLOCATIONS

GRADIENT THEORY FOR PLASTICITY VIA HOMOGENIZATION OF DISCRETE DISLOCATIONS GRADIENT THEORY FOR PLASTICITY VIA HOMOGENIZATION OF DISCRETE DISLOCATIONS ADRIANA GARRONI, GIOVANNI LEONI, AND MARCELLO PONSIGLIONE Abstract. In this paper, we deduce a macroscopic strain gradient theory

More information

Point topological defects in ordered media in dimension two

Point topological defects in ordered media in dimension two Point topological defects in ordered media in dimension two David CHIRON Laboratoire J.A. DIEUDONNE, Université de Nice - Sophia Antipolis, Parc Valrose, 68 Nice Cedex, France e-mail : chiron@math.unice.fr

More information

REGULARITY OF THE MINIMIZER FOR THE D-WAVE GINZBURG-LANDAU ENERGY

REGULARITY OF THE MINIMIZER FOR THE D-WAVE GINZBURG-LANDAU ENERGY METHODS AND APPLICATIONS OF ANALYSIS. c 2003 International Press Vol. 0, No., pp. 08 096, March 2003 005 REGULARITY OF THE MINIMIZER FOR THE D-WAVE GINZBURG-LANDAU ENERGY TAI-CHIA LIN AND LIHE WANG Abstract.

More information

Flat convergence of Jacobians and -convergence for the Ginzburg-Landau functionals by Sisto Baldo

Flat convergence of Jacobians and -convergence for the Ginzburg-Landau functionals by Sisto Baldo Lecture Notes of Seminario Interdisciplinare di Matematica Vol. 2 (2003), pp. 9-29. Flat convergence of Jacobians and -convergence for the Ginzburg-Landau functionals by Sisto Baldo Abstract. I try to

More information

Lorentz Space Estimates for the Ginzburg-Landau Energy

Lorentz Space Estimates for the Ginzburg-Landau Energy Lorentz Space Estimates for the Ginzburg-Landau Energy Sylvia Serfaty and Ian Tice Courant Institute of Mathematical Sciences 5 Mercer St., New York, NY serfaty@cims.nyu.edu, tice@cims.nyu.edu May, 7 Abstract

More information

On Moving Ginzburg-Landau Vortices

On Moving Ginzburg-Landau Vortices communications in analysis and geometry Volume, Number 5, 85-99, 004 On Moving Ginzburg-Landau Vortices Changyou Wang In this note, we establish a quantization property for the heat equation of Ginzburg-Landau

More information

On the distance between homotopy classes of maps between spheres

On the distance between homotopy classes of maps between spheres On the distance between homotopy classes of maps between spheres Shay Levi and Itai Shafrir February 18, 214 Department of Mathematics, Technion - I.I.T., 32 Haifa, ISRAEL Dedicated with great respect

More information

Integral Jensen inequality

Integral Jensen inequality Integral Jensen inequality Let us consider a convex set R d, and a convex function f : (, + ]. For any x,..., x n and λ,..., λ n with n λ i =, we have () f( n λ ix i ) n λ if(x i ). For a R d, let δ a

More information

SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction

SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy

More information

On Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations

On Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations On Smoothness of Suitable Weak Solutions to the Navier-Stokes Equations G. Seregin, V. Šverák Dedicated to Vsevolod Alexeevich Solonnikov Abstract We prove two sufficient conditions for local regularity

More information

Mean Field Limits for Ginzburg-Landau Vortices

Mean Field Limits for Ginzburg-Landau Vortices Mean Field Limits for Ginzburg-Landau Vortices Sylvia Serfaty Université P. et M. Curie Paris 6, Laboratoire Jacques-Louis Lions & Institut Universitaire de France Jean-Michel Coron s 60th birthday, June

More information

Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow

Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow Part I: Study of the perturbed Ginzburg-Landau equation Sylvia Serfaty Courant Institute of Mathematical Sciences 5 Mercer

More information

A Ginzburg-Landau Type Problem for Nematics with Highly Anisotropic Elastic Term

A Ginzburg-Landau Type Problem for Nematics with Highly Anisotropic Elastic Term A Ginzburg-Landau Type Problem for Nematics with Highly Anisotropic Elastic Term Peter Sternberg In collaboration with Dmitry Golovaty (Akron) and Raghav Venkatraman (Indiana) Department of Mathematics

More information

Lebesgue-Stieltjes measures and the play operator

Lebesgue-Stieltjes measures and the play operator Lebesgue-Stieltjes measures and the play operator Vincenzo Recupero Politecnico di Torino, Dipartimento di Matematica Corso Duca degli Abruzzi, 24, 10129 Torino - Italy E-mail: vincenzo.recupero@polito.it

More information

arxiv: v1 [math.fa] 26 Jan 2017

arxiv: v1 [math.fa] 26 Jan 2017 WEAK APPROXIMATION BY BOUNDED SOBOLEV MAPS WITH VALUES INTO COMPLETE MANIFOLDS PIERRE BOUSQUET, AUGUSTO C. PONCE, AND JEAN VAN SCHAFTINGEN arxiv:1701.07627v1 [math.fa] 26 Jan 2017 Abstract. We have recently

More information

SHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction

SHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms

More information

REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID

REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID REMARKS ON THE VANISHING OBSTACLE LIMIT FOR A 3D VISCOUS INCOMPRESSIBLE FLUID DRAGOŞ IFTIMIE AND JAMES P. KELLIHER Abstract. In [Math. Ann. 336 (2006), 449-489] the authors consider the two dimensional

More information

THE L 2 -HODGE THEORY AND REPRESENTATION ON R n

THE L 2 -HODGE THEORY AND REPRESENTATION ON R n THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some

More information

Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement

Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement Etienne Sandier and Sylvia Serfaty November 8, 00 Abstract We prove some improved estimates for the Ginzburg-Landau energy with

More information

Vortex collisions and energy-dissipation rates in the Ginzburg Landau heat flow

Vortex collisions and energy-dissipation rates in the Ginzburg Landau heat flow J. Eur. Math. Soc. 9, 177 17 c European Mathematical Society 007 Sylvia Serfaty Vortex collisions and energy-dissipation rates in the Ginzburg Landau heat flow Part I: Study of the perturbed Ginzburg Landau

More information

GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS

GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS LE MATEMATICHE Vol. LI (1996) Fasc. II, pp. 335347 GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS CARLO SBORDONE Dedicated to Professor Francesco Guglielmino on his 7th birthday W

More information

l(y j ) = 0 for all y j (1)

l(y j ) = 0 for all y j (1) Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that

More information

A generic property of families of Lagrangian systems

A generic property of families of Lagrangian systems Annals of Mathematics, 167 (2008), 1099 1108 A generic property of families of Lagrangian systems By Patrick Bernard and Gonzalo Contreras * Abstract We prove that a generic Lagrangian has finitely many

More information

ESTIMATES FOR THE MONGE-AMPERE EQUATION

ESTIMATES FOR THE MONGE-AMPERE EQUATION GLOBAL W 2,p ESTIMATES FOR THE MONGE-AMPERE EQUATION O. SAVIN Abstract. We use a localization property of boundary sections for solutions to the Monge-Ampere equation obtain global W 2,p estimates under

More information

CAMILLO DE LELLIS, FRANCESCO GHIRALDIN

CAMILLO DE LELLIS, FRANCESCO GHIRALDIN AN EXTENSION OF MÜLLER S IDENTITY Det = det CAMILLO DE LELLIS, FRANCESCO GHIRALDIN Abstract. In this paper we study the pointwise characterization of the distributional Jacobian of BnV maps. After recalling

More information

A NOTE ON ZERO SETS OF FRACTIONAL SOBOLEV FUNCTIONS WITH NEGATIVE POWER OF INTEGRABILITY. 1. Introduction

A NOTE ON ZERO SETS OF FRACTIONAL SOBOLEV FUNCTIONS WITH NEGATIVE POWER OF INTEGRABILITY. 1. Introduction A NOTE ON ZERO SETS OF FRACTIONAL SOBOLEV FUNCTIONS WITH NEGATIVE POWER OF INTEGRABILITY ARMIN SCHIKORRA Abstract. We extend a Poincaré-type inequality for functions with large zero-sets by Jiang and Lin

More information

ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang

ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS. Tian Ma. Shouhong Wang DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Volume 11, Number 1, July 004 pp. 189 04 ASYMPTOTIC STRUCTURE FOR SOLUTIONS OF THE NAVIER STOKES EQUATIONS Tian Ma Department of

More information

On the Intrinsic Differentiability Theorem of Gromov-Schoen

On the Intrinsic Differentiability Theorem of Gromov-Schoen On the Intrinsic Differentiability Theorem of Gromov-Schoen Georgios Daskalopoulos Brown University daskal@math.brown.edu Chikako Mese 2 Johns Hopkins University cmese@math.jhu.edu Abstract In this note,

More information

MATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Spring 2018 Professor: Jared Speck

MATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Spring 2018 Professor: Jared Speck MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Spring 208 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations

More information

On uniqueness of weak solutions to transport equation with non-smooth velocity field

On uniqueness of weak solutions to transport equation with non-smooth velocity field On uniqueness of weak solutions to transport equation with non-smooth velocity field Paolo Bonicatto Abstract Given a bounded, autonomous vector field b: R d R d, we study the uniqueness of bounded solutions

More information

NEARLY PARALLEL VORTEX FILAMENTS IN THE 3D GINZBURG-LANDAU EQUATIONS

NEARLY PARALLEL VORTEX FILAMENTS IN THE 3D GINZBURG-LANDAU EQUATIONS NEARLY PARALLEL VORTEX FILAMENTS IN THE 3D GINZBURG-LANDAU EQUATIONS ANDRES CONTRERAS AND ROBERT L. JERRARD arxiv:66.732v2 [math.ap] 6 Sep 27 Abstract. We introduce a framework to study the occurrence

More information

Lorentz space estimates for the Coulombian renormalized energy

Lorentz space estimates for the Coulombian renormalized energy Lorentz space estimates for the Coulombian renormalized energy Sylvia Serfaty and Ian Tice March 7, 0 Abstract In this paper we obtain optimal estimates for the currents associated to point masses in the

More information

Γ-convergence of functionals on divergence-free fields

Γ-convergence of functionals on divergence-free fields Γ-convergence of functionals on divergence-free fields N. Ansini Section de Mathématiques EPFL 05 Lausanne Switzerland A. Garroni Dip. di Matematica Univ. di Roma La Sapienza P.le A. Moro 2 0085 Rome,

More information

REGULARITY FOR INFINITY HARMONIC FUNCTIONS IN TWO DIMENSIONS

REGULARITY FOR INFINITY HARMONIC FUNCTIONS IN TWO DIMENSIONS C,α REGULARITY FOR INFINITY HARMONIC FUNCTIONS IN TWO DIMENSIONS LAWRENCE C. EVANS AND OVIDIU SAVIN Abstract. We propose a new method for showing C,α regularity for solutions of the infinity Laplacian

More information

EXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS

EXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS EXISTECE AD REGULARITY RESULTS FOR SOME OLIEAR PARABOLIC EUATIOS Lucio BOCCARDO 1 Andrea DALL AGLIO 2 Thierry GALLOUËT3 Luigi ORSIA 1 Abstract We prove summability results for the solutions of nonlinear

More information

Linear and Nonlinear Aspects of Vortices : the Ginzburg-Landau Model. F. Pacard T. Rivière

Linear and Nonlinear Aspects of Vortices : the Ginzburg-Landau Model. F. Pacard T. Rivière Linear and Nonlinear Aspects of Vortices : the Ginzburg-Landau Model F. Pacard T. Rivière January 28, 2004 Contents 1 Qualitative Aspects of Ginzburg-Landau Equations 1 1.1 The integrable case..........................

More information

PARTIAL REGULARITY OF BRENIER SOLUTIONS OF THE MONGE-AMPÈRE EQUATION

PARTIAL REGULARITY OF BRENIER SOLUTIONS OF THE MONGE-AMPÈRE EQUATION PARTIAL REGULARITY OF BRENIER SOLUTIONS OF THE MONGE-AMPÈRE EQUATION ALESSIO FIGALLI AND YOUNG-HEON KIM Abstract. Given Ω, Λ R n two bounded open sets, and f and g two probability densities concentrated

More information

AW -Convergence and Well-Posedness of Non Convex Functions

AW -Convergence and Well-Posedness of Non Convex Functions Journal of Convex Analysis Volume 10 (2003), No. 2, 351 364 AW -Convergence Well-Posedness of Non Convex Functions Silvia Villa DIMA, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy villa@dima.unige.it

More information

AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS. To the memory of our friend and colleague Fuensanta Andreu

AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS. To the memory of our friend and colleague Fuensanta Andreu AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Abstract. We characterize p-harmonic functions in terms of an asymptotic mean value

More information

Branched transport limit of the Ginzburg-Landau functional

Branched transport limit of the Ginzburg-Landau functional Branched transport limit of the Ginzburg-Landau functional Michael Goldman CNRS, LJLL, Paris 7 Joint work with S. Conti, F. Otto and S. Serfaty Introduction Superconductivity was first observed by Onnes

More information

MATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Fall 2011 Professor: Jared Speck

MATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Fall 2011 Professor: Jared Speck MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Fall 20 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations

More information

Note on the Chen-Lin Result with the Li-Zhang Method

Note on the Chen-Lin Result with the Li-Zhang Method J. Math. Sci. Univ. Tokyo 18 (2011), 429 439. Note on the Chen-Lin Result with the Li-Zhang Method By Samy Skander Bahoura Abstract. We give a new proof of the Chen-Lin result with the method of moving

More information

Metric Spaces and Topology

Metric Spaces and Topology Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies

More information

Global minimizers for a p-ginzburg-landau-type energy in R 2

Global minimizers for a p-ginzburg-landau-type energy in R 2 Global minimizers for a p-ginzburg-landau-type energy in R 2 Yaniv Almog, Leonid Berlyand, Dmitry Golovaty and Itai Shafrir Abstract Given a p > 2, we prove existence of global minimizers for a p-ginzburg-

More information

JUHA KINNUNEN. Harmonic Analysis

JUHA KINNUNEN. Harmonic Analysis JUHA KINNUNEN Harmonic Analysis Department of Mathematics and Systems Analysis, Aalto University 27 Contents Calderón-Zygmund decomposition. Dyadic subcubes of a cube.........................2 Dyadic cubes

More information

Probability and Measure

Probability and Measure Part II Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 84 Paper 4, Section II 26J Let (X, A) be a measurable space. Let T : X X be a measurable map, and µ a probability

More information

On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. Ben Schweizer 1

On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. Ben Schweizer 1 On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma Ben Schweizer 1 January 16, 2017 Abstract: We study connections between four different types of results that

More information

Calculus of Variations. Final Examination

Calculus of Variations. Final Examination Université Paris-Saclay M AMS and Optimization January 18th, 018 Calculus of Variations Final Examination Duration : 3h ; all kind of paper documents (notes, books...) are authorized. The total score of

More information

MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN

MULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN Electronic Journal of Differential Equations, Vol. 016 (016), No. 97, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIONS

More information

Line energy Ginzburg Landau models: zero energy states

Line energy Ginzburg Landau models: zero energy states Line energy Ginzburg Landau models: zero energy states Pierre-Emmanuel Jabin*, email: jabin@dma.ens.fr Felix Otto** email: otto@iam.uni-bonn.de Benoît Perthame* email: benoit.perthame@ens.fr *Département

More information

Compactness in Ginzburg-Landau energy by kinetic averaging

Compactness in Ginzburg-Landau energy by kinetic averaging Compactness in Ginzburg-Landau energy by kinetic averaging Pierre-Emmanuel Jabin École Normale Supérieure de Paris AND Benoît Perthame École Normale Supérieure de Paris Abstract We consider a Ginzburg-Landau

More information

Lecture 12. F o s, (1.1) F t := s>t

Lecture 12. F o s, (1.1) F t := s>t Lecture 12 1 Brownian motion: the Markov property Let C := C(0, ), R) be the space of continuous functions mapping from 0, ) to R, in which a Brownian motion (B t ) t 0 almost surely takes its value. Let

More information

Periodic solutions of weakly coupled superlinear systems

Periodic solutions of weakly coupled superlinear systems Periodic solutions of weakly coupled superlinear systems Alessandro Fonda and Andrea Sfecci Abstract By the use of a higher dimensional version of the Poincaré Birkhoff theorem, we are able to generalize

More information

Liquid crystal flows in two dimensions

Liquid crystal flows in two dimensions Liquid crystal flows in two dimensions Fanghua Lin Junyu Lin Changyou Wang Abstract The paper is concerned with a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of

More information

02. Measure and integral. 1. Borel-measurable functions and pointwise limits

02. Measure and integral. 1. Borel-measurable functions and pointwise limits (October 3, 2017) 02. Measure and integral Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2017-18/02 measure and integral.pdf]

More information

From the Brunn-Minkowski inequality to a class of Poincaré type inequalities

From the Brunn-Minkowski inequality to a class of Poincaré type inequalities arxiv:math/0703584v1 [math.fa] 20 Mar 2007 From the Brunn-Minkowski inequality to a class of Poincaré type inequalities Andrea Colesanti Abstract We present an argument which leads from the Brunn-Minkowski

More information

TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS. Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017

TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS. Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017 TOPICS IN NONLINEAR ANALYSIS AND APPLICATIONS Dipartimento di Matematica e Applicazioni Università di Milano Bicocca March 15-16, 2017 Abstracts of the talks Spectral stability under removal of small capacity

More information

An introduction to Mathematical Theory of Control

An introduction to Mathematical Theory of Control An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018

More information

Laplace s Equation. Chapter Mean Value Formulas

Laplace s Equation. Chapter Mean Value Formulas Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic

More information

MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW

MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW GREGORY DRUGAN AND XUAN HIEN NGUYEN Abstract. We present two initial graphs over the entire R n, n 2 for which the mean curvature flow

More information

The Lusin Theorem and Horizontal Graphs in the Heisenberg Group

The Lusin Theorem and Horizontal Graphs in the Heisenberg Group Analysis and Geometry in Metric Spaces Research Article DOI: 10.2478/agms-2013-0008 AGMS 2013 295-301 The Lusin Theorem and Horizontal Graphs in the Heisenberg Group Abstract In this paper we prove that

More information

NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES

NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES JUHA LEHRBÄCK Abstract. We establish necessary conditions for domains Ω R n which admit the pointwise (p, β)-hardy inequality u(x) Cd Ω(x)

More information

Math The Laplacian. 1 Green s Identities, Fundamental Solution

Math The Laplacian. 1 Green s Identities, Fundamental Solution Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external

More information

Two-dimensional multiple-valued Dirichlet minimizing functions

Two-dimensional multiple-valued Dirichlet minimizing functions Two-dimensional multiple-valued Dirichlet minimizing functions Wei Zhu July 6, 2007 Abstract We give some remarks on two-dimensional multiple-valued Dirichlet minimizing functions, including frequency,

More information

Weak Topologies, Reflexivity, Adjoint operators

Weak Topologies, Reflexivity, Adjoint operators Chapter 2 Weak Topologies, Reflexivity, Adjoint operators 2.1 Topological vector spaces and locally convex spaces Definition 2.1.1. [Topological Vector Spaces and Locally convex Spaces] Let E be a vector

More information

AN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS. Vieri Benci Donato Fortunato. 1. Introduction

AN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS. Vieri Benci Donato Fortunato. 1. Introduction Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume, 998, 83 93 AN EIGENVALUE PROBLEM FOR THE SCHRÖDINGER MAXWELL EQUATIONS Vieri Benci Donato Fortunato Dedicated to

More information

BIHARMONIC WAVE MAPS INTO SPHERES

BIHARMONIC WAVE MAPS INTO SPHERES BIHARMONIC WAVE MAPS INTO SPHERES SEBASTIAN HERR, TOBIAS LAMM, AND ROLAND SCHNAUBELT Abstract. A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed.

More information

On the distributional divergence of vector fields vanishing at infinity

On the distributional divergence of vector fields vanishing at infinity Proceedings of the Royal Society of Edinburgh, 141A, 65 76, 2011 On the distributional divergence of vector fields vanishing at infinity Thierry De Pauw Institut de Recherches en Mathématiques et Physique,

More information

Nonlinear Energy Forms and Lipschitz Spaces on the Koch Curve

Nonlinear Energy Forms and Lipschitz Spaces on the Koch Curve Journal of Convex Analysis Volume 9 (2002), No. 1, 245 257 Nonlinear Energy Forms and Lipschitz Spaces on the Koch Curve Raffaela Capitanelli Dipartimento di Metodi e Modelli Matematici per le Scienze

More information

(1.3) G m (h ex ) def. (1.4) G ε (u ε, A ε ) = G m + (π ln ε + 2πd j h ex ξ m (a j )) + o ε ( ln ε ), (1.5) u ε (x) ρ ε ( x a j )

(1.3) G m (h ex ) def. (1.4) G ε (u ε, A ε ) = G m + (π ln ε + 2πd j h ex ξ m (a j )) + o ε ( ln ε ), (1.5) u ε (x) ρ ε ( x a j ) A MODEL FOR VORTEX NUCLEATION IN THE GINZBURG-LANDAU EQUATIONS GAUTAM IYER 1 AND DANIEL SPIRN 2 Abstract. This paper studies questions related to the dynamic transition between local and global minimizers

More information

EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018

EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018 EXPOSITORY NOTES ON DISTRIBUTION THEORY, FALL 2018 While these notes are under construction, I expect there will be many typos. The main reference for this is volume 1 of Hörmander, The analysis of liner

More information

Euler Equations: local existence

Euler Equations: local existence Euler Equations: local existence Mat 529, Lesson 2. 1 Active scalars formulation We start with a lemma. Lemma 1. Assume that w is a magnetization variable, i.e. t w + u w + ( u) w = 0. If u = Pw then u

More information

Analysis in weighted spaces : preliminary version

Analysis in weighted spaces : preliminary version Analysis in weighted spaces : preliminary version Frank Pacard To cite this version: Frank Pacard. Analysis in weighted spaces : preliminary version. 3rd cycle. Téhéran (Iran, 2006, pp.75.

More information

14 Divergence, flux, Laplacian

14 Divergence, flux, Laplacian Tel Aviv University, 204/5 Analysis-III,IV 240 4 Divergence, flux, Laplacian 4a What is the problem................ 240 4b Integral of derivative (again)........... 242 4c Divergence and flux.................

More information

Introduction to Real Analysis Alternative Chapter 1

Introduction to Real Analysis Alternative Chapter 1 Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces

More information

Improved estimates for the Ginzburg-Landau equation: the elliptic case

Improved estimates for the Ginzburg-Landau equation: the elliptic case Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. IV (2005), 319-355 Improved estimates for the Ginzburg-Landau equation: the elliptic case FABRICE BETHUEL, GIANDOMENICO ORLANDI AND DIDIER SMETS Abstract.

More information

AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS

AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS AN ASYMPTOTIC MEAN VALUE CHARACTERIZATION FOR p-harmonic FUNCTIONS JUAN J. MANFREDI, MIKKO PARVIAINEN, AND JULIO D. ROSSI Abstract. We characterize p-harmonic functions in terms of an asymptotic mean value

More information

MINIMAL SURFACES AND MINIMIZERS OF THE GINZBURG-LANDAU ENERGY

MINIMAL SURFACES AND MINIMIZERS OF THE GINZBURG-LANDAU ENERGY MINIMAL SURFACES AND MINIMIZERS OF THE GINZBURG-LANDAU ENERGY O. SAVIN 1. Introduction In this expository article we describe various properties in parallel for minimal surfaces and minimizers of the Ginzburg-Landau

More information

Takens embedding theorem for infinite-dimensional dynamical systems

Takens embedding theorem for infinite-dimensional dynamical systems Takens embedding theorem for infinite-dimensional dynamical systems James C. Robinson Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. E-mail: jcr@maths.warwick.ac.uk Abstract. Takens

More information

Geometric Measure Theory

Geometric Measure Theory Encyclopedia of Mathematical Physics, Vol. 2, pp. 520 527 (J.-P. Françoise et al. eds.). Elsevier, Oxford 2006. [version: May 19, 2007] 1. Introduction Geometric Measure Theory Giovanni Alberti Dipartimento

More information

Mean Field Limits for Ginzburg-Landau Vortices

Mean Field Limits for Ginzburg-Landau Vortices Mean Field Limits for Ginzburg-Landau Vortices Sylvia Serfaty Courant Institute, NYU Rivière-Fabes symposium, April 28-29, 2017 The Ginzburg-Landau equations u : Ω R 2 C u = u ε 2 (1 u 2 ) Ginzburg-Landau

More information

Behaviour of Lipschitz functions on negligible sets. Non-differentiability in R. Outline

Behaviour of Lipschitz functions on negligible sets. Non-differentiability in R. Outline Behaviour of Lipschitz functions on negligible sets G. Alberti 1 M. Csörnyei 2 D. Preiss 3 1 Università di Pisa 2 University College London 3 University of Warwick Lars Ahlfors Centennial Celebration Helsinki,

More information

Introduction to Bases in Banach Spaces

Introduction to Bases in Banach Spaces Introduction to Bases in Banach Spaces Matt Daws June 5, 2005 Abstract We introduce the notion of Schauder bases in Banach spaces, aiming to be able to give a statement of, and make sense of, the Gowers

More information

MATH 411 NOTES (UNDER CONSTRUCTION)

MATH 411 NOTES (UNDER CONSTRUCTION) MATH 411 NOTES (NDE CONSTCTION 1. Notes on compact sets. This is similar to ideas you learned in Math 410, except open sets had not yet been defined. Definition 1.1. K n is compact if for every covering

More information

A LOWER BOUND FOR THE GRADIENT OF -HARMONIC FUNCTIONS Edi Rosset. 1. Introduction. u xi u xj u xi x j

A LOWER BOUND FOR THE GRADIENT OF -HARMONIC FUNCTIONS Edi Rosset. 1. Introduction. u xi u xj u xi x j Electronic Journal of Differential Equations, Vol. 1996(1996) No. 0, pp. 1 7. ISSN 107-6691. URL: http://ejde.math.swt.edu (147.6.103.110) telnet (login: ejde), ftp, and gopher access: ejde.math.swt.edu

More information

An introduction to semilinear elliptic equations

An introduction to semilinear elliptic equations An introduction to semilinear elliptic equations Thierry Cazenave Laboratoire Jacques-Louis Lions UMR CNRS 7598 B.C. 187 Université Pierre et Marie Curie 4, place Jussieu 75252 Paris Cedex 05 France E-mail

More information

An introduction to Birkhoff normal form

An introduction to Birkhoff normal form An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an

More information

An introduction to semilinear elliptic equations

An introduction to semilinear elliptic equations An introduction to semilinear elliptic equations Thierry Cazenave Laboratoire Jacques-Louis Lions UMR CNRS 7598 B.C. 187 Université Pierre et Marie Curie 4, place Jussieu 75252 Paris Cedex 05 France E-mail

More information

On the validity of the Euler Lagrange equation

On the validity of the Euler Lagrange equation J. Math. Anal. Appl. 304 (2005) 356 369 www.elsevier.com/locate/jmaa On the validity of the Euler Lagrange equation A. Ferriero, E.M. Marchini Dipartimento di Matematica e Applicazioni, Università degli

More information