Small energy regularity for a fractional Ginzburg-Landau system - PowerPoint PPT Presentation

Small energy regularity for a fractional Ginzburg-Landau system Yannick Sire University Aix-Marseille Work in progress with Vincent Millot (Univ. Paris 7) mercredi 6 juin 2012

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 � v ( x ) − v ( y ) � γ N := Γ(( N + 1) / 2) � ( − ∆) 1 / 2 v ( x ) := PV γ N | x − y | N +1 dy π ( N +1) / 2 R N for smooth bounded functions v • We eventually complement the equation with a “exterior” Dirichlet con- dition in R N \ ω v = g for a given (smooth) bounded function g mercredi 6 juin 2012

Functional setting - Variational formulation loc ( R N ; R M ) ∩ L ∞ we can define ( − ∆) 1 / 2 v in D ′ ( ω ) by • For v ∈ H 1 / 2 ( v ( x ) − v ( y )) · ( ϕ ( x ) − ϕ ( y )) �� := γ N � � ( − ∆) 1 / 2 v, ϕ dxdy | x − y | N +1 2 ω × ω ( v ( x ) − v ( y )) · ( ϕ ( x ) − ϕ ( y )) �� + γ N dxdy | x − y | N +1 ω × ( R N \ ω ) • Conclusion 1: ( − ∆) 1 / 2 is related to the first variation (in ω ) of | v ( x ) − v ( y ) | 2 | v ( x ) − v ( y ) | 2 �� �� E ( v, ω ) := γ N dxdy + γ N dxdy | x − y | N +1 | x − y | N +1 4 2 ω × ω ω × ( R N \ ω ) mercredi 6 juin 2012

• 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 ( ω ), the 00 dual space of H 1 / 2 00 ( ω ), with � ( − ∆) 1 / 2 v � H − 1 / 2 � ( ω ) ≤ E ( v, ω ) 00 • Conclusion 3: Variational formulation of the (FGL) system. We look at variational solutions of (FGL), i.e., critical points (w.r.t. per- turbations in ω ) of the fractional Ginzburg-Landau energy E ε ( v, ω ) := E ( v, ω ) + 1 � (1 − | v | 2 ) 2 dx 4 ε ω In other words, we are interested in solutions of � d � for all ϕ ∈ H 1 / 2 dt E ε ( v + tϕ, ω ) = 0 00 ( ω ) t =0 • Minimizing solutions under Dirichlet condition: the easiest way to find such solutions is to solve the minimization problem E ε ( v, ω ) : v ∈ g + H 1 / 2 � � min 00 ( ω ) for a (smooth) bounded function g : R N → R M mercredi 6 juin 2012

Main goal - Motivations • Extend recent results to the vectorial setting. Allen-Cahn equation with fractional diffusion: 1. Alberti - Bouchitt´ e - Seppecher 2. Cabr´ e - Sol` a Morales 3. Garroni - Palatucci 4. Sire - Valdinoci 5. Savin - Valdinoci 6. .... • Half-harmonic maps into spheres: Da Lio & T. Rivi` ere Regularity of critical points v : R → S M − 1 of � | ( − ∆) 1 / 4 v | 2 dx I ( v ) := R ⇒ v ∈ C ∞ ( R ) (analogue of H´ elein ’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. mercredi 6 juin 2012

• 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, allow- ing for (entire) local minimizers, critical points in bounded domains with ”Dirichlet” condition, etc ... (slightly di ff erent 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  in R N \ B 1 v = g  ⇒ 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. mercredi 6 juin 2012

Half-harmonic maps into spheres Definition: loc ( R N ; R M ) ∩ L ∞ be such that | v | = 1 a.e. in ω . We shall say that Let v ∈ H 1 / 2 v is a weak half-harmonic map into S M − 1 in ω if � d � v + tϕ �� dt E = 0 | v + tϕ | t =0 00 ( ω ) ∩ L ∞ compactly supported in ω . for all ϕ ∈ H 1 / 2 Euler-Lagrange equations: loc ( R N ; R M ) ∩ L ∞ such that | v | = 1 a.e. in ω is weakly half- A map v ∈ H 1 / 2 harmonic 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 ( ω ) 00 mercredi 6 juin 2012

Half-Laplacian Vs Dirichlet-to-Neumann operator • Harmonic extension - Poisson Formula: := R N × (0 , + ∞ ), For v defined on R N , we set for x = ( x ′ , x N +1 ) ∈ R N +1 + x N +1 v ( y ′ ) � v ext ( x ) := γ N dy ′ ( | x ′ − y ′ | 2 + x 2 N +1 N +1 ) R N 2 • 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 1 � � |∇ v ext | 2 dx = min |∇ u | 2 dx : u = v on ∂ R N +1 ∼ R N + 2 2 R N +1 R N +1 + + Moreover,  ∆ v ext = 0 in R N +1  + v ext = v on ∂ R N +1 ∼ R N  + mercredi 6 juin 2012

• Harmonic extension for H 1 / 2 loc -functions: For v ∈ H 1 / 2 loc ( R N ) ∩ 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 ( R N ) ∩ L ∞ , we have � � � ∇ v ext · ∇ Φ dx ∀ ϕ ∈ H 1 / 2 ( − ∆) 1 / 2 v, ϕ = 00 ( ω ) , R N +1 + N +1 where Φ ∈ H 1 ( R N +1 and Φ | R N = ϕ ) is compactly supported in R + + • 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 |∇ u | 2 dx − 1 � � |∇ v ext | 2 dx ≥ E ( u | R N , ω ) − E ( v, ω ) 2 2 Ω Ω for all u ∈ H 1 (Ω) such that u − v ext = 0 in a neighborhood of ∂ Ω \ ω mercredi 6 juin 2012

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 = ∂v ext loc ( R N ) ∩ L ∞ is a solution , if v ∈ H 1 / 2 ∂ν of the (FGL) system in ω , then its harmonic extension v ext solves  ∆ u = 0 in Ω   ∂ν = 1 ∂u 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 | ∇ u | 2 dx + 1 � � (1 − | u | 2 ) 2 dx 2 4 ε ω Ω mercredi 6 juin 2012

• Minimizing solutions for (FGL): loc ( R N ) ∩ L ∞ is a minimizing solution of (FGL) We shall say that v ∈ H 1 / 2 in ω if E ε ( v, ω ) ≤ E ε (˜ v, ω ) v ∈ H 1 / 2 loc ( R N ) such that ˜ for all ˜ v − 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 (˜ u, Ω) u ∈ H 1 (Ω) such that ˜ for all ˜ u − u is compactly supported in Ω ∪ ω . Comparison between Fractional and Dirichlet energy: loc ( R N ) ∩ L ∞ is a minimizing solution of (FGL) in ω , then v ext is a If v ∈ H 1 / 2 minimizing solution of (GLB) in Ω. mercredi 6 juin 2012

• Interior regularity for (GLB): ( Cabr´ e & Sola Morales ) If u ∈ H 1 (Ω) ∩ L ∞ solves  ∆ u = 0 in Ω   ∂ν = 1 ∂u ε (1 − | u | 2 ) u on ω ,   then u ∈ C ∞ (Ω ∪ ω ). � x N +1 Trick: consider w ( x ) := u ( x ′ , t ) dt 0 • 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): loc ( R N ) ∩ L ∞ solves (FGL) in ω , then v ∈ C ∞ ( ω ). If v ∈ H 1 / 2 If v satisfies in addition, u = g on R N \ ω for a smooth bounded function g , then v is H¨ older continuous accross ∂ω . mercredi 6 juin 2012

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 in H − 1 / 2 ∂ν ⊥ T u S M − 1 ( ω )   00 mercredi 6 juin 2012