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 : RN → RM of the system

(−∆)1/2v = 1 ε(1 − |v|2)v in ω , where ε > 0 is a small, and ω is (smooth) bounded open subset of RN

• The integro-differential operator (−∆)1/2 is defined by

(−∆)1/2v(x) := PV

• γN
• RN

v(x) − v(y) |x − y|N+1 dy

• γN := Γ((N + 1)/2)

π(N+1)/2 for smooth bounded functions v

• We eventually complement the equation with a “exterior” Dirichlet con-

dition v = g in RN \ ω for a given (smooth) bounded function g

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Functional setting - Variational formulation

• For v ∈ H1/2

loc (RN; RM) ∩ L∞ we can define (−∆)1/2v in D′(ω) by

• (−∆)1/2v, ϕ
• := γN

2

• ω×ω

(v(x) − v(y)) · (ϕ(x) − ϕ(y)) |x − y|N+1 dxdy + γN

• ω×(RN\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) := γN 4

• ω×ω

|v(x) − v(y)|2 |x − y|N+1 dxdy + γN 2

• ω×(RN\ω)

|v(x) − v(y)|2 |x − y|N+1 dxdy

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• Conclusion 2: Actually we can define (−∆)1/2v whenever E(v, ω) < ∞

(which holds for v ∈ H1/2

loc ∩ L∞), and then (−∆)1/2v ∈ H−1/2 00

(ω), the dual space of H1/2

00 (ω), with

(−∆)1/2vH−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. per- turbations in ω) of the fractional Ginzburg-Landau energy Eε(v, ω) := E(v, ω) + 1 4ε

• ω

(1 − |v|2)2 dx In other words, we are interested in solutions of d dtEε(v + tϕ, ω)

• t=0

= 0 for all ϕ ∈ H1/2

00 (ω)

• Minimizing solutions under Dirichlet condition: the easiest way to

find such solutions is to solve the minimization problem min

• Eε(v, ω) : v ∈ g + H1/2

00 (ω)

• for a (smooth) bounded function g : RN → RM

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 → SM−1 of I(v) :=

• R

|(−∆)1/4v|2 dx ⇒ 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 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    (−∆)1/2v = 1 ε(1 − |v|2)v in B1 v = g in RN \ B1 ⇒ 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.

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Half-harmonic maps into spheres

Definition: Let v ∈ H1/2

loc (RN; RM) ∩ L∞ be such that |v| = 1 a.e. in ω. We shall say that

v is a weak half-harmonic map into SM−1 in ω if d dtE v + tϕ |v + tϕ|

• t=0

= 0 for all ϕ ∈ H1/2

00 (ω) ∩ L∞ compactly supported in ω.

Euler-Lagrange equations: A map v ∈ H1/2

loc (RN; RM) ∩ L∞ such that |v| = 1 a.e. in ω is weakly half-

harmonic in ω if

• (−∆)1/2v, ϕ
• = 0

for all ϕ ∈ H1/2

00 (ω) satisfying

ϕ(x) ∈ Tv(x)SM−1 a.e. in ω Or equivalently, (−∆v)1/2 ⊥ TvSM−1 in H−1/2

00

(ω)

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Half-Laplacian Vs Dirichlet-to-Neumann operator

• Harmonic extension - Poisson Formula:

For v defined on RN, we set for x = (x′, xN+1) ∈ RN+1

+

:= RN × (0, +∞), vext(x) := γN

• RN

xN+1 v(y′) (|x′ − y′|2 + x2

N+1)

N+1 2

dy′

• Entire fractional energy Vs Dirichlet energy:

For v ∈ H1/2(RN) it is well known that vext ∈ H1(RN), and E(v, RN) = 1 2

• RN+1

+

|∇vext|2 dx = min

• 1

2

• RN+1

+

|∇u|2 dx : u = v on ∂RN+1

+

∼ RN

• Moreover,

   ∆vext = 0 in RN+1

+

vext = v

• n ∂RN+1

+

∼ RN

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• Harmonic extension for H1/2

loc -functions:

For v ∈ H1/2

loc (RN) ∩ L∞, we have E(v, Br) < ∞ for all r > 0, and the

harmonic extension vext is still well defined with vext ∈ H1

loc(RN+1 +

) ∩ L∞

• The half-Laplacian as a Dirichlet-to-Neumann operator:

For v ∈ H1/2

loc (RN) ∩ L∞, we have

• (−∆)1/2v, ϕ
• =
• RN+1

+

∇vext · ∇Φ dx ∀ϕ ∈ H1/2

00 (ω) ,

where Φ ∈ H1(RN+1

+

) is compactly supported in R

N+1 +

and Φ|RN = ϕ

• Fractional energy Vs Dirichlet energy: ∼(Caffarelli-Roquejoffre-Savin)

Let Ω ⊂ RN+1

+

be a bounded Lipschitz open set such that ω ⊂ ∂Ω. Then, 1 2

• Ω

|∇u|2 dx − 1 2

• Ω

|∇vext|2 dx ≥ E(u|RN , ω) − E(v, ω) for all u ∈ H1(Ω) such that u − vext = 0 in a neighborhood of ∂Ω \ ω

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System of semi-linear boundary reactions

Let Ω ⊂ RN+1

+

be a bounded Lipschitz open set such that ω ⊂ ∂Ω. By the charactization (−∆)1/2v = ∂vext ∂ν , if v ∈ H1/2

loc (RN)∩L∞ is a solution

• f the (FGL) system in ω, then its harmonic extension vext solves

     ∆u = 0 in Ω ∂u ∂ν = 1 ε(1 − |u|2)u

• n ω

In conclusion: To study the (FBL) system, it suffices to consider this system

• f “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 Eε(u, Ω) := 1 2

• Ω

|∇u|2 dx + 1 4ε

• ω

(1 − |u|2)2 dx

mercredi 6 juin 2012

• Minimizing solutions for (FGL):

We shall say that v ∈ H1/2

loc (RN) ∩ L∞ is a minimizing solution of (FGL)

in ω if Eε(v, ω) ≤ Eε(˜ v, ω) for all ˜ v ∈ H1/2

loc (RN) such that ˜

v − v is compactly supported in ω.

• Minimizing solutions for (GLB):

We shall say that u ∈ H1(Ω) is a minimizing solution of (GLB) in Ω if Eε(u, Ω) ≤ Eε(˜ u, Ω) for all ˜ u ∈ H1(Ω) such that ˜ u − u is compactly supported in Ω ∪ ω. Comparison between Fractional and Dirichlet energy: If v ∈ H1/2

loc (RN) ∩ L∞ is a minimizing solution of (FGL) in ω, then vext is a

minimizing solution of (GLB) in Ω.

mercredi 6 juin 2012

• Interior regularity for (GLB): (Cabr´

e & Sola Morales) If u ∈ H1(Ω) ∩ L∞ solves      ∆u = 0 in Ω ∂u ∂ν = 1 ε(1 − |u|2)u

• n ω ,

then u ∈ C∞(Ω ∪ ω). Trick: consider w(x) := xN+1 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 ∈ Cβ(Ω).

• Consequence for (FGL):

If v ∈ H1/2

loc (RN) ∩ L∞ solves (FGL) in ω, then v ∈ C∞(ω).

If v satisfies in addition, u = g on RN \ ω for a smooth bounded function g, then v is H¨

• lder continuous accross ∂ω.

mercredi 6 juin 2012

Boundary harmonic maps into spheres

Let Ω ⊂ RN+1

+

be a bounded Lipschitz open set such that ω ⊂ ∂Ω. Definition of (weak) Boundary harmonic map: Let u ∈ H1(Ω; RM) be such that |u|∂Ω| = 1 a.e. in ω. We shall say that u is a weak boundary harmonic map into SM−1 in (Ω, ω) if

• Ω

∇u · ∇Φ dx = 0 for all Φ ∈ H1(Ω; RM) ∩ L∞ compactly supported in Ω ∪ ω and satisfying Φ(x) ∈ Tu(x)SM−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 ⊥ TuSM−1 in H−1/2

00

(ω)

mercredi 6 juin 2012

Remarks: 1) the definition is motivated by the fact that d dt 1 2

• Ω

|∇ut|2 dx

• t=0

=

• Ω

∇u · ∇Φ dx for variations ut of the form ut := 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 SM−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 op- erator, if v ∈ H1/2

loc (RN) ∩ L∞ is a (weak) half-harmonic map into SM−1 in ω,

then vext is a (weak) boundary harmonic map into SM−1 in (Ω, ω).

mercredi 6 juin 2012

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 ∈ H1(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

|∇˜ u|2 dx for all ˜ u ∈ H1(Ω) such that |˜ u|∂Ω| = 1 a.e. in ω, and ˜ u − u is compactly supported in Ω ∪ ω.

• Minimizing half-harmonic maps:

We shall say that v ∈ H1/2

loc (RN) ∩ L∞ satisfying |v| = 1 a.e. in ω, is a

minimizing half-harmonic map in ω if E(v, ω) ≤ E(˜ v, ω) for all ˜ v ∈ H1/2

loc (RN) such that |˜

v| = 1 a.e. in ω, and ˜ v − v is compactly supported in ω.

mercredi 6 juin 2012

Comparison between Fractional and Dirichlet energy: If v ∈ H1/2

loc (RN) ∩ L∞ is a minimizing half-harmonic map in ω, then vext is a

minimizing boundary harmonic map in (Ω, ω). Stationnary boundary harmonic maps: We shall say that u ∈ H1(Ω) satisfying |u|∂Ω| = 1 a.e. in ω, is a stationnary boundary harmonic map in Ω if d dt 1 2

• Ω

|∇(u ◦ φt)|2 dx

• t=0

= 0 for all smooth 1-parameter families of C∞-diffeomorphism φt : Ω → Ω satisfying

• 1. φ0 = idΩ
• 2. φt(∂Ω ∩ {xN+1 = 0}) ⊂ ∂Ω ∩ {xN+1 = 0}
• 3. φt − idΩ is compactly supported in Ω ∪ ω

“Stationnary” half-harmonic maps: We shall say that v ∈ H1/2

loc (RN) ∩ L∞ satisfying |v| = 1 a.e. in ω, is a “station-

nary” half-harmonic map in ω if vext is a stationnary boundary harmonic map in (Ω, ω).

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Theorem 1. (Scheven) If N = 1 and u ∈ H1(Ω) ∩ L∞ is weak boundary harmonic map in SM−1 in (Ω, ω), then u ∈ C∞(Ω ∪ ω). Theorem 2. (Scheven) Let N ≥ 2 and assume that u ∈ H1(Ω) ∩ L∞ is a stationnary boundary harmonic map in SM−1 in (Ω, ω). Then there exists a relatively closed set Σ ⊂ ω such that HN−1(Σ) = 0 and u ∈ C∞ Ω ∪ (ω \ Σ)

• .

Theorem 3. (Duzaar & Steffen, Hardt & Lin) Let N ≥ 2 and assume that u ∈ H1(Ω) ∩ L∞ is a minimizing boundary harmonic map in SM−1 in (Ω, ω). Then there exists a relatively closed set Σ ⊂ ω such that dimH(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 SM−1 2) For the mixed boundary value problem, boundary regularity at the edge ∂ω is not known (Duzaar & Grotowski)

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“Small energy” gradient-estimate for (GLB)

For x ∈ ∂RN+1

+

, set B+

R(x) := BR(x) ∩ RN+1 +

and DR(x) := BR(x) ∩ ∂RN+1

+

Theorem. Let R > 0 and ε > 0. There exist constants η0 > 0 and C0 > 0 (indep. of R and ε) such that for each map u ∈ C1(B

+ R) satisfying |u| ≤ 1 and

     ∆u = 0 in B+

R ,

∂u ∂ν = 1 ε(1 − |u|2)u

• n DR ,

the condition 1 RN−1 Eε(u, B+

R) ≤ η0 ,

implies sup

B+

R/4

|∇u|2 + sup

DR/4

(1 − |u|2)2 ε2 ≤ C0 R2 η0

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Application 1: bounded energy solutions of (GLB)

Theorem. Let εn ↓ 0. For each n ∈ N, let un ∈ H1(Ω) be a solution of      ∆un = 0 in Ω , ∂un ∂ν = 1 εn (1 − |un|2)un

• n ω ,

such that |un| ≤ 1 and supn Eεn(un, Ω) < ∞. Then there exist a subsequence and a weak boundary harmonic map u∗ into SM−1 in (Ω, ω) such that un ⇀ u∗ weakly in H1(Ω). In addition, there exist a non-negative Radon measure µ in ω, and a relatively closed set Σ ⊂ ω of locally finite HN−1-measure such that (i) 1 2|∇un|2 dx + 1 4εn (1 − |un|2)2 dx′ ∗ ⇀ 1 2|∇u∗|2 dx + µ as measures; (ii) Σ = spt(µ) ∪ sing(u∗) ; (iii) un → u∗ in C1,α

loc

• Ω ∪ (ω \ Σ)
• for every 0 < α < 1.

Finally, for N = 1 the set Σ is locally finite in ω.

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Application 2: minimizers of (FGL)

• Theorem. (M ≥ 2)

Let εn ↓ 0, and g : RN → RM a smooth function satisfying |g| = 1 in RN \ ω. For each n ∈ N, let vn ∈ argmin

• Eε(v, ω) : v ∈ g + H1/2

00 (ω)

• .

Then there exist a subsequence and a minimizing half-harmonic map v∗ into SM−1 in ω such that (vn − v∗) → 0 strongly in H1/2

00 (ω). In addition, vn → v∗

in C1,α

loc

• ω \ sing(v∗)
• for every 0 < α < 1.

Remarks: 1) The assumption M ≥ 2 ensures that

• v ∈ g + H1/2

00 (ω) : |v| = 1 a.e.

• = ∅

2) Example: N = M, ω = D1, and g(x) = x/|x| 3) Do we have smooth convergence near the boundary ∂ω ?

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Key ingredients for small energy regularity

By smoothness of solutions of (GLB), Stationnarity holds whence: Energy Monotonicity Formula: Let u ∈ C1(B

+ R) solving

     ∆u = 0 in B+

R ,

∂u ∂ν = 1 ε(1 − |u|2)u

• n DR .

Then for every x ∈ DR and 0 < ρ < r < dist(x, ∂DR), 1 ρN−1 Eε(u, B+

ρ (x)) ≤

1 rN−1 Eε(u, B+

r (x))

Remark: Liouville type property. The only finite energy entire solutions of (GBL) or (FGL) are constant functions

mercredi 6 juin 2012

In the spirit of Stationnary harmonic maps with a free boundary (Scheven) Clearing-out lemma: For 0 < ε < 1, there exists a η1 > 0 (indep. of ε) such that for each map u ∈ C1(B

+ 1 ) satisfying |u| ≤ 1 and

     ∆u = 0 in B+

1 ,

∂u ∂ν = 1 ε(1 − |u|2)u

• n D1 ,

the condition Eε(u, B+

1 ) ≤ η1 ,

implies |u| ≥ 1/2 in B+

1/2

Consequence. We can use the use polar decomposition u = aw with a = |u| and w = u |u|

mercredi 6 juin 2012

Polar decomposition of (GLB): Setting a = |u| and u = aw (assuming a ≥ 1/2), we have      −∆a + |∇w|2a = 0 in B+

1

∂a ∂ν = 1 ε(1 − a2)a

• n D1

,      −div(a2∇w) = a2|∇w|2w in B+

1

∂w ∂ν = 0

• n D1

Small energy regularity, Strategy: 1) Blow-up around a “high energy” point (localisation “` a la Chen-Struwe”) 2) Prove compactness in C1,α for solutions bounded in C1 and in energy Limiting system:    −∆a∗ + |∇w∗|2a∗ = 0 in B+

1

a∗ = 1

• n D1

,      −div(a2

∗∇w∗) = a2|∇w∗|2w∗

in B+

1

∂w∗ ∂ν = 0

• n D1

mercredi 6 juin 2012

Construction of super-solutions: 1) For the standard Ginzburg-Landau system:    −ε2∆w + w = 0 in B1 w = 1

• n ∂B1

⇒ w is exponentially small in ε in B1/2 2) For the Boundary Ginzburg-Landau system:              −∆w = 1 in B+

1

w = 1

• n ∂B+

1 ∩ {xN+1 > 0}

ε∂w ∂ν + w = 0

• n D1

⇒ w is linearly small in ε in D1/2

mercredi 6 juin 2012