Small energy regularity for a fractional Ginzburg-Landau system

Transcription

1 Small energy regularity for a fractional Ginzburg-Landau system Yannick Sire University Aix-Marseille Work in progress with Vincent Millot (Univ. Paris 7)

2 The fractional Ginzburg-Landau system We are interest in (weak) bounded solutions v : R N R M of the system ( ) 1/2 v = 1 ε (1 v 2 )v in ω, where ε > 0 is a small, and ω is (smooth) bounded open subset of R N The integro-differential operator ( ) 1/2 is defined by ) ( ) 1/2 v(x) v(y) v(x) := P V (γ N dy γ x y N+1 N := R N Γ((N + 1)/2) π (N+1)/2 for smooth bounded functions v We eventually complement the equation with a exterior Dirichlet condition v = g in R N \ ω for a given (smooth) bounded function g

3 Functional setting - Variational formulation For v H 1/2 loc (RN ; R M ) L we can define ( ) 1/2 v in D (ω) by ( ) 1/2 v, ϕ := γ N 2 ω ω (v(x) v(y)) (ϕ(x) ϕ(y)) x y N+1 dxdy + γ N ω (R N \ω) (v(x) v(y)) (ϕ(x) ϕ(y)) x y N+1 dxdy Conclusion 1: ( ) 1/2 is related to the first variation (in ω) of E(v, ω) := γ N 4 ω ω v(x) v(y) 2 x y N+1 dxdy + γ N 2 ω (R N \ω) v(x) v(y) 2 x y N+1 dxdy

4 Conclusion 2: Actually we can define ( ) 1/2 v whenever E(v, ω) < (which holds for v H 1/2 loc L ), and then ( ) 1/2 v H 1/2 00 (ω), the dual space of H 1/2 00 (ω), with ( ) 1/2 v H 1/2 00 (ω) E(v, ω) Conclusion 3: Variational formulation of the (FGL) system. We look at variational solutions of (FGL), i.e., critical points (w.r.t. perturbations in ω) of the fractional Ginzburg-Landau energy E ε (v, ω) := E(v, ω) + 1 (1 v 2 ) 2 dx 4ε In other words, we are interested in solutions of ω [ ] d dt E ε(v + tϕ, ω) t=0 = 0 for all ϕ H 1/2 00 (ω) Minimizing solutions under Dirichlet condition: the easiest way to find such solutions is to solve the minimization problem min { E ε (v, ω) : v g + H 1/2 00 (ω)} for a (smooth) bounded function g : R N R M

5 Main goal - Motivations Extend recent results to the vectorial setting. Allen-Cahn equation with fractional diffusion: 1. Alberti - Bouchitté - Seppecher 2. Cabré - Solà Morales 3. Garroni - Palatucci 4. Sire - Valdinoci 5. Savin - Valdinoci Half-harmonic maps into spheres: Da Lio & T. Rivière Regularity of critical points v : R S M 1 of I(v) := ( ) 1/4 v 2 dx R v C (R) (analogue of Hélein s result on weak harmonic maps in 2D) In their paper, they suggest that half-harmonic maps arise as limits of the (FGL) system as ε 0.

6 Find a useful localized energy for half-harmonic maps: Liouville type theorem: In higher dimensions, entire half-harmonic maps with finite energy are trivial!! We ve been looking for a localized version of the problem, allowing for (entire) local minimizers, critical points in bounded domains with Dirichlet condition, etc... (slightly different approach by Moser) Research program: Extend the results of F.H. Lin & C. Wang to the fractional setting (for GL, related to the blow-up analysis of harmonic maps by F.H. Lin) A model case: For N = M 2, take g(x) = x/ x, and solve ( ) 1/2 v = 1 ε (1 v 2 )v in B 1 v = g in R N \ B 1 as ε 0, we should have v 1. On the other hand, g does not admit a continuous extension of modulus one by standard degree theory.

7 Half-harmonic maps into spheres Definition: Let v H 1/2 loc (RN ; R M ) L be such that v = 1 a.e. in ω. We shall say that v is a weak half-harmonic map into S M 1 in ω if [ d dt E ( v + tϕ v + tϕ )] t=0 = 0 for all ϕ H 1/2 00 (ω) L compactly supported in ω. Euler-Lagrange equations: A map v H 1/2 loc (RN ; R M ) L such that v = 1 a.e. in ω is weakly halfharmonic in ω if ( ) 1/2 v, ϕ = 0 for all ϕ H 1/2 00 (ω) satisfying ϕ(x) T v(x) S M 1 a.e. in ω Or equivalently, ( v) 1/2 T v S M 1 in H 1/2 00 (ω)

8 Half-Laplacian Vs Dirichlet-to-Neumann operator Harmonic extension - Poisson Formula: For v defined on R N, we set for x = (x, x N+1 ) R N+1 + := R N (0, + ), v ext (x) := γ N x N+1 v(y ) R N ( x y 2 + x 2 N+1 ) N+1 2 dy Entire fractional energy Vs Dirichlet energy: For v H 1/2 (R N ) it is well known that v ext H 1 (R N ), and E(v, R N ) = 1 2 R N+1 + v ext 2 dx = min { 1 2 R N+1 + u 2 dx : u = v on R N+1 + R N } Moreover, v ext = 0 in R N+1 + v ext = v on R N+1 + R N

9 Harmonic extension for H 1/2 loc -functions: For v H 1/2 loc (RN ) L, we have E(v, B r ) < for all r > 0, and the harmonic extension v ext is still well defined with v ext H 1 loc(r N+1 + ) L The half-laplacian as a Dirichlet-to-Neumann operator: For v H 1/2 loc (RN ) L, we have ( ) 1/2 v, ϕ = R N+1 + v ext Φ dx ϕ H 1/2 00 (ω), where Φ H 1 (R N+1 + ) is compactly supported in R N+1 + and Φ R N = ϕ Fractional energy Vs Dirichlet energy: (Caffarelli-Roquejoffre-Savin) Let Ω R N+1 + be a bounded Lipschitz open set such that ω Ω. Then, 1 2 Ω u 2 dx 1 2 Ω v ext 2 dx E(u R N, ω) E(v, ω) for all u H 1 (Ω) such that u v ext = 0 in a neighborhood of Ω \ ω

10 System of semi-linear boundary reactions Let Ω R N+1 + be a bounded Lipschitz open set such that ω Ω. By the charactization ( ) 1/2 v = vext, if v H 1/2 loc ν (RN ) L is a solution of the (FGL) system in ω, then its harmonic extension v ext solves u = 0 in Ω u ν = 1 ε (1 u 2 )u on ω In conclusion: To study the (FBL) system, it suffices to consider this system of boundary reactions : Ginzburg-Landau Boundary System (GLB) The Ginzburg-Landau (boundary) energy: Solution of (GLB) correspond to critical points (w.r.t. compactly supported pertutbations in Ω ω) of the energy E ε (u, Ω) := 1 2 Ω u 2 dx + 1 4ε ω (1 u 2 ) 2 dx

11 Minimizing solutions for (FGL): We shall say that v H 1/2 loc (RN ) L is a minimizing solution of (FGL) in ω if E ε (v, ω) E ε (ṽ, ω) for all ṽ H 1/2 loc (RN ) such that ṽ v is compactly supported in ω. Minimizing solutions for (GLB): We shall say that u H 1 (Ω) is a minimizing solution of (GLB) in Ω if E ε (u, Ω) E ε (ũ, Ω) for all ũ H 1 (Ω) such that ũ u is compactly supported in Ω ω. Comparison between Fractional and Dirichlet energy: If v H 1/2 loc (RN ) L is a minimizing solution of (FGL) in ω, then v ext is a minimizing solution of (GLB) in Ω.

12 Interior regularity for (GLB): (Cabré & Sola Morales) If u H 1 (Ω) L solves u = 0 in Ω u ν = 1 ε (1 u 2 )u on ω, then u C (Ω ω). Trick: consider w(x) := x N+1 0 u(x, t) dt Boundary (edge) regularity for (GLB) under Dirichlet condition: If u satisfies in addition, u = g on Ω \ ω for a smooth function g, then u C β (Ω). Consequence for (FGL): If v H 1/2 loc (RN ) L solves (FGL) in ω, then v C (ω). If v satisfies in addition, u = g on R N \ ω for a smooth bounded function g, then v is Hölder continuous accross ω.

13 Boundary harmonic maps into spheres Let Ω R N+1 + be a bounded Lipschitz open set such that ω Ω. Definition of (weak) Boundary harmonic map: Let u H 1 (Ω; R M ) be such that u Ω = 1 a.e. in ω. We shall say that u is a weak boundary harmonic map into S M 1 in (Ω, ω) if u Φ dx = 0 Ω for all Φ H 1 (Ω; R M ) L compactly supported in Ω ω and satisfying Φ(x) T u(x) S M 1 a.e. in ω Equivalently: Choosing Φ with compact support in Ω shows that u is harmonic in Ω. Integrating by parts allows to rephrase the definition as u = 0 in Ω u ν T us M 1 in H 1/2 00 (ω)

14 Remarks: 1) the definition is motivated by the fact that [ ( )] d 1 u t 2 dx = dt 2 for variations u t of the form Ω t=0 Ω u Φ dx u t := u + tφ 1 + t2 Φ 2 with Φ as above 2) for bounded solutions, boundary harmonic maps belong to the class of Harmonic maps with Free Boundary where ω is the free boundary and S M 1 is the supporting manifold. Duzaar & Steffen, Duzaar & Grotowski, Hardt & Lin, Scheven,... Half-harmonic map Vs Boundary harmonic map: By the characterization of ( ) 1/2 in terms of the Dirichlet-to-Neumann operator, if v H 1/2 loc (RN ) L is a (weak) half-harmonic map into S M 1 in ω, then v ext is a (weak) boundary harmonic map into S M 1 in (Ω, ω).

15 Consequence: In general there is no hope of regularity or partial regularity for weak boundary harmonic maps. to have partial regularity, we should consider minimizing or stationnary boundary harmonic maps Minimizing boundary harmonic maps: We shall say that u H 1 (Ω) satisfying u Ω = 1 a.e. in ω, is a minimizing boundary harmonic map in (Ω, ω) if 1 2 Ω u 2 dx 1 2 Ω ũ 2 dx for all ũ H 1 (Ω) such that ũ Ω = 1 a.e. in ω, and ũ u is compactly supported in Ω ω. Minimizing half-harmonic maps: We shall say that v H 1/2 loc (RN ) L satisfying v = 1 a.e. in ω, is a minimizing half-harmonic map in ω if E(v, ω) E(ṽ, ω) for all ṽ H 1/2 loc (RN ) such that ṽ = 1 a.e. in ω, and ṽ v is compactly supported in ω.

16 Comparison between Fractional and Dirichlet energy: If v H 1/2 loc (RN ) L is a minimizing half-harmonic map in ω, then v ext is a minimizing boundary harmonic map in (Ω, ω). Stationnary boundary harmonic maps: We shall say that u H 1 (Ω) satisfying u Ω = 1 a.e. in ω, is a stationnary boundary harmonic map in Ω if [ ( )] d 1 (u φ t ) 2 dx = 0 dt 2 Ω t=0 for all smooth 1-parameter families of C -diffeomorphism φ t : Ω Ω satisfying 1. φ 0 = id Ω 2. φ t ( Ω {x N+1 = 0}) Ω {x N+1 = 0} 3. φ t id Ω is compactly supported in Ω ω Stationnary half-harmonic maps: We shall say that v H 1/2 loc (RN ) L satisfying v = 1 a.e. in ω, is a stationnary half-harmonic map in ω if v ext is a stationnary boundary harmonic map in (Ω, ω).

17 Theorem 1. (Scheven) If N = 1 and u H 1 (Ω) L is weak boundary harmonic map in S M 1 in (Ω, ω), then u C (Ω ω). Theorem 2. (Scheven) Let N 2 and assume that u H 1 (Ω) L is a stationnary boundary harmonic map in S M 1 in (Ω, ω). Then there exists a relatively closed set Σ ω such that H N 1 (Σ) = 0 and u C ( Ω (ω \ Σ) ). Theorem 3. (Duzaar & Steffen, Hardt & Lin) Let N 2 and assume that u H 1 (Ω) L is a minimizing boundary harmonic map in S M 1 in (Ω, ω). Then there exists a relatively closed set Σ ω such that dim H (Σ) N 2 if N 3, Σ is discrete if N = 2, and u C ( Ω (ω \ Σ) ). Remarks 1) Same statements for half-harmonic maps into S M 1 2) For the mixed boundary value problem, boundary regularity at the edge ω is not known (Duzaar & Grotowski)

18 Small energy gradient-estimate for (GLB) For x R N+1 +, set B + R (x) := B R(x) R N+1 + and D R (x) := B R (x) R N+1 + Theorem. Let R > 0 and ε > 0. There exist constants η 0 > 0 and C 0 > 0 (indep. of R and ε) such that for each map u C 1 (B + R) satisfying u 1 and u = 0 in B + R, the condition implies sup B + R/4 u ν = 1 ε (1 u 2 )u on D R, 1 R N 1 E ε(u, B + R ) η 0, u 2 + sup D R/4 (1 u 2 ) 2 ε 2 C 0 R 2 η 0

19 Application 1: bounded energy solutions of (GLB) Theorem. Let ε n 0. For each n N, let u n H 1 (Ω) be a solution of u n = 0 in Ω, u n ν = 1 ε n (1 u n 2 )u n on ω, such that u n 1 and sup n E εn (u n, Ω) <. Then there exist a subsequence and a weak boundary harmonic map u into S M 1 in (Ω, ω) such that u n u weakly in H 1 (Ω). In addition, there exist a non-negative Radon measure µ in ω, and a relatively closed set Σ ω of locally finite H N 1 -measure such that (i) 1 2 u n 2 dx + 1 4ε n (1 u n 2 ) 2 dx 1 2 u 2 dx + µ as measures; (ii) Σ = spt(µ) sing(u ) ; (iii) u n u in C 1,α ( ) loc Ω (ω \ Σ) for every 0 < α < 1. Finally, for N = 1 the set Σ is locally finite in ω.

20 Application 2: minimizers of (FGL) Theorem. (M 2) Let ε n 0, and g : R N R M a smooth function satisfying g = 1 in R N \ ω. For each n N, let v n argmin { E ε (v, ω) : v g + H 1/2 00 (ω)}. Then there exist a subsequence and a minimizing half-harmonic map v into S M 1 in ω such that (v n v ) 0 strongly in H 1/2 00 (ω). In addition, v n v in C 1,α ( loc ω \ sing(v ) ) for every 0 < α < 1. Remarks: 1) The assumption M 2 ensures that { v g + H 1/2 00 (ω) : v = 1 a.e. } 2) Example: N = M, ω = D 1, and g(x) = x/ x 3) Do we have smooth convergence near the boundary ω?

21 Key ingredients for small energy regularity By smoothness of solutions of (GLB), Stationnarity holds whence: Energy Monotonicity Formula: Let u C 1 (B + R) solving u = 0 in B + R, u ν = 1 ε (1 u 2 )u on D R. Then for every x D R and 0 < ρ < r < dist(x, D R ), 1 ρ N 1 E ε(u, B ρ + (x)) 1 r N 1 E ε(u, B r + (x)) Remark: Liouville type property. The only finite energy entire solutions of (GBL) or (F GL) are constant functions

22 In the spirit of Stationnary harmonic maps with a free boundary (Scheven) Clearing-out lemma: For 0 < ε < 1, there exists a η 1 > 0 (indep. of ε) such that for each map u C 1 (B + 1 ) satisfying u 1 and u = 0 in B 1 +, u ν = 1 ε (1 u 2 )u on D 1, the condition implies E ε (u, B + 1 ) η 1, u 1/2 in B + 1/2 Consequence. We can use the use polar decomposition u = aw with a = u and w = u u

23 Polar decomposition of (GLB): Setting a = u and u = aw (assuming a 1/2), we have a + w 2 a = 0 in B 1 + div(a 2 w) = a 2 w 2 w in B 1 + a ν = 1, ε (1 a2 )a on D 1 w ν = 0 on D 1 Small energy regularity, Strategy: 1) Blow-up around a high energy point (localisation à la Chen-Struwe ) 2) Prove compactness in C 1,α for solutions bounded in C 1 and in energy Limiting system: a + w 2 a = 0 in B 1 +, a = 1 on D 1 div(a 2 w ) = a 2 w 2 w in B 1 + w ν = 0 on D 1

24 Construction of super-solutions: 1) For the standard Ginzburg-Landau system: ε 2 w + w = 0 in B 1 w = 1 on B 1 w is exponentially small in ε in B 1/2 2) For the Boundary Ginzburg-Landau system: w = 1 in B 1 + w = 1 on B 1 + {x N+1 > 0} ε w ν + w = 0 on D 1 w is linearly small in ε in D 1/2