Interaction energy between vortices of vector fields on Riemannian surfaces

Transcription

1 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 Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

2 We consider 3 related problems for vector fields on 2-dimensounal Riemannian manifolds: Problem 1: Intrinsic Let (S, g) be a compact 2-dimensional Riemannian manifold. Consider tangent vector fields u, and minimize the intrinsic energy Iε in (u) = 1 [ Du 2 g + 1 ] 1 u 2 2 S ε 2 g vol g. Here Du 2 g(x) := D τ1 u 2 g(x) + D τ2 u 2 g(x) where D v denotes covariant differentiation and {τ 1, τ 2 } are any orthonormal basis for T x S. Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

3 Problem 2: extrinsic, tangent vector fields Let (S, g) be a compact, connected, oriented 2-dimensional Riemannian manifold isometrically embedded in R 3. Consider sections m of the tangent bundle of S, and minimize the extrinsic energy Iε ex (m) = 1 [ 2 Dm ] 1 m 2 S ε 2 dh 2 Here m H 1 (S; R 3 ), with m(x) T x S for every x S, and Dm 2 := D τ1 m 2 + D τ2 m 2, where m is an extension of m to a neighborhood of S, {τ 1 (x), τ 2 (x)} form a basis for T x S, D v denotes covariant derivative in R 3. well known: Dm 2 is independent of the choice of extension m. Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

4 Problem 3: extrinsic, S 2 constraint Let (S, g) be compact a 2-dimensional Riemannian manifold isometrically embedded in R 3. Consider maps M : S S 2, and minimize Iε S2 (M) = 1 [ M 2 + 1ε ] 2 2 (M ν)2 dh 2 S Here M 2 := τ 1 M 2 + τ 2 M 2, where M is an extension of S to a neighborhood of S and {τ 1 (x), τ 2 (x)} form a basis for T x S. As usual, M 2 is independent of the choice of extension M Remark 1: If M = 1 and m denotes the tangential part of M, then (M ν) 2 = 1 m 2 = 1 m 2. Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

5 Remark 2 : Let S R 3 be a fixed smooth surface isometrically embedded in R 3. A curved magnetic shell is considered occupying the domain Ω h := { x + sn(x ) : s (0, h), x S }. The magnetization m : Ω h S 2 is a stable state of the energy functional E 3D (m) = ε Ω 2 m 2 + U 2 dx, h R 3 where U : R 3 R solves the static Maxwell equation ) U = (m1 Ω in R 3. Carbou (2001) shows that Iε S2 and h 0. arises as the Γ-limit of E 3D with ε fixed This is the original motivation for our study. Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

6 example Iε in (u left ) = 0, Iε in (u right ) = 0, Iε ex (u left ) = 0, Iε ex (u right ) > 0, I S2 ε (u left ) = 0, I S2 ε (u right ) > 0. Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

7 related problems simplified Ginzburg-Landau on a manifold (S, g) abstract manifold, ψ H 1 (S; C), I ε (ψ) := 1 ψ ε 2 (1 ψ 2 ) 2 d vol g. S See Baraket (1996). (Compare Bethuel-Brezis-Hélein (1994) for Euclidean case) Ginzburg-Landau on a complex line bundle ψ a section of a complex line bundle E over a Riemann surface S. A a connection on E. G ε (ψ, A) := 1 D A ψ 2 + F A ε 2 (1 ψ 2 ) 2 dh 2. S See Orlandi (1996), Qing (1997). (Compare Bethuel-Rivière (1994) for Euclidean case) Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

8 Ginzburg-Landau on thin shells (S, g) isometrically embedded in R 3, ψ H 1 (S; C), I ε (ψ) := 1 ( i(a e ) τ )ψ ε 2 (1 ψ 2 ) 2 dh 2. S See Contreras-Sternberg (2010), Contreras (2011). Related work Alama-Bronsard-Galvao-Sousa (2010, 2013) (Compare Sandier-Serfaty (late 90s) for Euclidean case) discrete-to-continuum limit (S, g) isometrically embedded in R 3, T ε := ε- triangulation of S, I disc ε (ψ) := 1 2 i j T ε κ ij ε ψ ε (i) ψ ε (j) 2, where ψ ε (i) T i S, ψ ε (i) = 1 for all i. Canevari-Segatti (2017) Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

9 prior results: Euler characteristic nonzero lim inf ε 0 I ε = +. Canevari, Segatti, Veneroni (2015), Segatti, Snarski, Veneroni (2016) study of variational problem when Euler characteristic = 0. Segatti, Snarski, Veneroni (2016) New results (Ignat - J 2017) For Problems 1-3, canonical unit vector fields and renormalized energy for prescribed singularities and fluxes, as in Bethuel-Brezis-Hélein (1994). in every case, second-order Γ-convergence". Extrinsic Problem 2 (tangent constraint) and Problem 3 (S 2 constraint with penalization) have essentially the same asymptotics. Problem 1 (intrinsic) admits a lifting to a linear problem (with topological considerations). Problems 2 and 3 seem to be inescapably nonlinear. intrinsic canonical harmonic unit vector field provides Coulomb gauge for the more nonlinear Problems 2,3. Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

10 general set-up for intrinsic problem always assume S is oriented can then define i : TS TS such that i isometry of T x S to itself for every x, and {v, iv} properly oriented orthonormal basis of T x S, or every unit v T x S. given any vector field u, define 1-form j(u) by j(u)(v) = (D v u, iu) g define vorticity associated to u by ω(u) = dj(u) + κ vol g, κ = Gaussian curvature. Remark: if u is a smooth unit vector field in an open set O, then dj(u) = κ vol g and thus ω(u) = 0 in O. Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

11 canonical harmonic unit vector field Theorem For any a 1,..., a k S and d 1,..., d k Z such that d k = χ(s), there exists unit vector field u in W 1,p for all p < 2, such that ω(u ) = 2π d i δ ai, d j(u ) = 0. If g := genus(s) = 0, then u is unique up to a global phase. If g := genus(s) > 0, then u is unique up to a global phase, once 2g flux integrals" Φ l, l = 1,..., 2g are specified. Finally, L(a, d) := {admissible values of (Φ 1,..., Φ 2g )} are quantized and depend smoothly on d i δ ai (All results: Ignat - J, 2017) Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

12 outline of proof Given a i, d i as above 1 Find 1-form j as described below 2 find unit vector field u such that j(u ) = j. If genus> 0 need to pay attention to topology. Construction of j. if j(u ) = j, then equations for u become dj = κ vol g + 2π d i δ ai, d j = 0. In fact where j = d ψ + harmonic 1-form, if g > 0 ψ = κ vol g + 2π d i δ ai The condition j = j(unit vector field) implies constraints on the harmonic 1-form. Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

13 Construction of u : To solve find u such that j(u ) = j : choose p S and v T p S, and set u (p) := v. given q S, consider γ : [0, 1] S with γ(0) = p, γ(1) = q. Let U(s) T γ(s) S solve D γ (s)u(s) = j(γ (s)) iu(s), U(0) = v T p S. Define u (q) = U(1). check that this is well-defined. This determines L(a, d). Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

14 intrinsic renormalized energy Theorem Given a i, d i as above, let u be the canonical harmonic unit vector field with flux integrals {Φ k } Then [ 1 lim r 0 2 Du 2 g ( ] di 2 )π log 1 = W in (a, d, Φ) r where S\ B(r,a i ) W in (a, d, Φ) = 4π 2 l k d l d k G(a l, a k ) + 2π n [ πd 2 k H(a k, a k ) + d k ψ 0 (a k ) ] k= Φ 2 + C S, where G(, ) is the Green s function for the Laplacian with regular part H(, ), and ψ 0 = κ + κ Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

15 extrinsic renormalized energy Theorem Suppose (S, g) is isometrically embedded in R 3. Let a i, d i be given such that d i = χ(s), and fix u = u (a, d, Φ). Suppose that u = e iα u for some α H 1 (S; R). Then for the extrinsic Dirichlet energy, W ex (a, d, Φ) := lim r 0 Here [ 1 S\ B(r,a i ) 2 Du 2 g ( ] di 2 )π log 1 r = W in (a, d, Φ) + S ( ) 1 2 α 2 g + Q u (cos α, sin α) vol g Q u (cos α, sin α) = A(e iα u ) 2, A = 2nd fundamental form is an explicit quadratic function of cos α, sin α. Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

16 Theorem (Γ-convergence) 1) (Compactness) Let (u ε ) ε 0 be a family of vector fields on S satisfying I ε (u ε ) Nπ log ε + C, = in, ex, S2. Then there exists a sequence ε 0 such that ω(u ε ) 2π n d k δ ak in W 1,1, as ε 0, k=1 where {a k } n k=1 are distinct points in S and {d k} n k=1 are nonzero integers satisfying d k = χ(s) and d k N. Moreover, if n k=1 d k = N, then n = N, d k = 1 for every k = 1,..., n and for a subsequence, there exists Φ L(a, d) such that ( ) Φ(u ε ) := (j(u ε ), η 1 ) g vol g,..., (j(u ε ), η 2g ) g vol g Φ S S as ε 0. (in particular, n = χ(s) modulo 2). Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

17 Theorem (Γ-convergence, continued) 2) (Γ-liminf inequality) Assume that the vector fields u ε X 1,2 (S) satisfy n ω(u ε ) 2π d k δ ak in W 1,1, as ε 0, k=1 Φ(u ε ) Φ L(a, d) (1) for n distinct points {a k } n k=1 Sn with d k = 1. Then [ ] lim inf Iε (u ε ) nπ log ε ) W (a, d, Φ) + nγ F. ε 0 3) (Γ-limsup inequality) For every n distinct points a 1,..., a n S and d 1,..., d n {±1} satisfying d k = χ(s) and every Φ L(a, d) there exists a sequence of vector fields u ε on S such that (1) holds and I ε (u ε ) nπ log ε W (a, d, Φ) + nγ F as ε 0. Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

18 Proofs use vortex ball construction indirect method in the Calculus of Variations: optimality/lower bounds follow (essentially) from equations that characterize u : ω(u ) = dj(u ) + κ vol g = 2π d j(u) = 0. careful accounting involving flux integrals Φ. n d k δ ak k=1 in Euclidean case, derivation of renormalized energy via 2nd-order Γ convergence: Colliander-J 1999, L i n - X i n 1999, Alicandro-Ponsiglione Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

19 the indirect method" From elementary algebra, e in 1 ε (u ε ) = S rε S rε 2 j ε 2 g + 1 j(u ε ) 2 j ε u ε g Here S rε = S \ B(a k,ε, r ε ). In fact the integrands are pointwise equal. In addition, as ε 0, 1 jε 2 vol = π( 2 S rε k 2 g + 2(j ε, j(u ε) u ε g j ε ) g + e in ε ( u ε g ) d 2 k ) log 1 r ε + W (a ε, d ε, Φ ε ) + O( r ε ) + O(r 2 ε Φ ε 2 ). So we only need to estimate S rε (j ε, j(u) u g j ε ) g vol g. Equations for j ε (with vortex ball construction) imply d(j(u) j ε ) j ε = d ψ ε + is small 2g k=1 Φ ε,k η k Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

20 thank you for your attention! Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

21 thank you for your attention! what s left of it, after a long day... Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

22 thank you for your attention! what s left of it, after a long day... Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21

23 thank you for your attention! what s left of it, after a long day... Ignat and Jerrard (To(ulouse,ronto) ) Ginzburg-Landau for vector fields 3/62/ / 21