Noncompact variational problems involving complex unimodular maps

Noncompact variational problems involving complex unimodular maps

Petru Mironescu

Institut Camille Jordan, Université Lyon 1

Taipei, October 25th, 2013

France-Taiwan Joint Conference on Nonlinear PDEs

Overview

∗ Digression: a compact problem

∗ The main tool: a minimum of the modulus principle

∗ A non compact variational problem

∗ Uniqueness

∗ Existence for largeε

∗ Existence for smallε

“Compact motivation”

Theorem (Dong Ye –Feng Zhou 96)

AssumeΩ⊂R²simply connected,g :∂Ω→S¹smooth of degree 0

Consider the Ginzburg-Landau type (GL) energy E_ε(u) = 1

2 Z

Ω

|∇u|²+ 1 4ε²

Z

Ω

(1− |u|²)²

subject tou=gon∂Ω

Then, for smallε, there is only one minimizeru_εofE_ε

If, in addition,kg−1k_C2 1, then uniqueness holds for everyε

Main lines of the proof

∗ For smallε,|u_ε| ≈1

∗ Ωbeing simply connected, writeu_ε=ρ_εe^ıϕε

∗ The equation ofu_εis “close” to the limiting equation∆ϕ_ε=0

∗ Uniqueness of the solution of the limiting

equation=⇒uniqueness of the solution of theε–equation

Technically...

∗ Proof relies on uniform (pointwise) bounds on∇ϕ_ε

∗ This is used to measure how “far” theε–problem is from the limiting problem (essential for the perturbative argument) However, natural assumption isg ∈H^1/2(∂Ω;S¹)

When we work withg∈H^1/2(∂Ω;S¹)maps...

∗ Mapsg ∈H^1/2(∂Ω;S¹)do have a degree (Boutet de Monvel, Gabber 91)

∗ For smallεand degg =0, we still have|u_ε| ≈1

∗ But the perturbative argument does not work anymore

Uniqueness in degree 0

Theorem (Farina, M 11)

AssumeΩsimply connected andg ∈H^1/2(∂Ω;S¹)of degree 0 Then, for smallε,E_ε has a unique minimizer

If, in addition,|g−1|_H1/2 1, thenE_εhas a unique minimizer for anyε

A basic ingredient

Minimum of the modulus principle in a nutshell

If an energy minimizeruhas small energy, then|u|is almost constant

Minimum of the modulus principle

LetuminimizeEεwrt its own Dirichlet bc in a simply connected domainΩ

If|u|=1 on∂Ωand|u|<ssomewhere inΩ, then Z

Ω

|∇u|²≥f(s)>0 (with explicitf)

Example

Whens=0, the argument givesf(0) =2π Which is optimal: takeε=∞,Ω =D,u =Id

Remarks

∗ εindependent conclusion

∗ Also works for minimizers of Z

Ω

|∇u|²+ Z

Ω

F(|u|), with suitableF. The conclusion isF-independent

∗ Does notwork in multiply connected domains (counterintuitive)

Proof of uniqueness for almost constant g

The proof of

“|g−1|_H1/2 1 =⇒ uniqueness of the mininimizer ofE_ε,∀ε”

relies on Wente estimates (Wente 69) Theorem (Bethuel, Ghidaglia 93)

Letu∈H₀¹(Ω)solve−∆u=∇f ∧ ∇g. Then kuk_L^∞ ≤2k∇fk_L2k∇gk_L2

k∇uk_L2 ≤√

2k∇fk_L2k∇gk_L2

Z

Ω

h∇f ∧ ∇g

≤√

2k∇fk_L2k∇gk_L2k∇hk_L2,∀h∈H₀¹(Ω)

Proof of uniqueness for almost constant g

and on...

Identity (Lassoued, M 99)

Letu,v 6=0, withu critical point ofE_ε. Writeu=ρe^ıϕ, v =uηe^ıψ. Then

Eε(v) =Eε(u)+≈ Z

Ω

|∇ψ|²

| {z }

good term

+≈ Z

|∇η|²

| {z }

good term

+

+ Z

Ω

(η²−1)ρ²∇ϕ· ∇ψ

| {z }

bad term

+≈ 1 ε²

Z

Ω

(1−η²)²

| {z }

good potential term

Strategy for uniqueness

∗ Letu,v be minimizers

∗ Prove thatu 6=0 andv 6=0

∗ Control the bad term in order to arrive atEε(v)−Eε(u)>0 except whenu=v

Proof of uniqueness for almost constant g

Sketch

∗ galmost constant=⇒ ∃an almost constant competitoru₀of modulus 1

∗ Thus minimal energy satisfies (1)E_ε(u_ε)≤E_ε(u₀)1

∗ By the minimum of the modulus principle, (2)|u_ε| −11

∗ GL equation =⇒the bad term can be rewritten as (3)

Z

Ω

(η²−1)ρ²∇ϕ· ∇ψ= Z

Ω

(η²−1)∇H∧ ∇ψ for someH

∗ (1) + (2) + (3) + Wente estimates=⇒ bad term is controlled by the good terms

Proof of asymptotic uniqueness for arbitrary g

The proof of

“degg =0 =⇒ uniqueness of the mininimizer ofEεfor smallε”

relies on “asymptotic Wente estimates”

Baby estimate

LetU_ε∈H₀¹(Ω)satisfy

−∆U_ε+ 1

ε²U_ε =∇f_ε∧ ∇g_ε If(f_ε)converges inH¹(Ω), then

k∇U_εk_L2 =o(1)k∇g_εk_L2

Proof of asymptotic uniqueness for arbitrary g

Sketch

∗ Prove that minimizers satisfy|u_ε| →1 asε→0 (blow up argument)

∗ Determine the equation satisfied byη²−1

∗ Use this equation + asymptotic Wente estimates to control the bad term via the good terms (surprise: in the final computation, no need of the good potential term)

A non compact problem: GL with semi-stiff BC

Semi-stiff GL problem

Find minimizers/critical points ofE_εin a (generally multiply connected) domainΩ⊂R²with bc:

|u|=1 on∂Ωand deg(u,Γ_j) =d_j given (withΓ_j component of

∂Ω)

Main features (partly conjectural)

∗ Allows boundary vortices (another model: Kurzke 06 for thin magnetic films)

∗ Critical points always exist (for smallε)

∗ Minimizers sometimes do exist, sometimes do not exist

∗ Non compact problem

Golovaty, Berlyand 02, Berlyand, M 06, Golovaty, Berlyand, Rybalko 09, Dos Santos 09, Berlyand, Rybalko 10, Farina, M 11, Berlyand, M, Sandier, Rybalko 12, Lamy, M 13

A existence + uniqueness case

Theorem (Golovaty, Berlyand 02)

LetΩ =DR\D, withR−11 (thin circular annulus) ThenE_εattains its minimum in the class

{u : Ω→C;|u|=1 on∂Ω,degu =1 onC_R andC₁} In addition, “the” minimizer is unique and radial

Uniqueness in thin domains

Theorem (Farina, M 11)

There is someδ >0 s.t., if infE_ε(u)< δ₀, thenE_ε has a

“unique” minimizer (with prescribed degrees) Remark

δ₀does not depend on the prescribed degrees

Sketch of proof

∗ Prove compactness of minimizing sequences. Otherwise, formation of bubbles. But not enough energy

∗ This leads to existence of minimizers

∗ Extend the minimum of the modulus principle to prescribed degrees minimizers

∗ Then proceed as for the uniqueness of minimizers in case of almost constantg

Remark

Recall that, in multiply connected domains, the minimum of the modulus principle requiressomeextra assumption in addition toubeing a minimizer ofE_εwrt its own Dirichlet bc

Existence issues

What is known

∗ In multiply connected domains, critical points do exist for smallε(Berlyad, Rybalko 10 for doubly connected domain, Dos Santos 09 for general multiply connected domains)

∗ Such solutions are built as local minimizers ofE_ε

∗ Construction does not work in simply connected domains;

similar with help from the topology of the domain in case of the critical Sobolev exponent (Coron 84, Bahri – Coron 88)

Existence issues

In simply connected domains

∗ Fact: no minimizer (except forε=∞)

∗ Critical points do exist (partial results)

Proposition

Ifε <∞and prescribed degree6=0, then no minimizer Proof.

Asume e.g.Ω =Dandd =1. Then|∇u|²≥2Jacu =⇒ 1

2 Z

D

|∇u|²≥ Z

D

Jacu=πdeg(u, ∂D) =π so thatE_ε(u)> π

Now testE_ε(M_a)(Moebius transform centered ata) and let

|a| %1 to obtain infEε(u)≤π Remark

In general, infE_ε=πd, withd the prescribed degree, and inf is not attained

Analysis for large ε

Theorem (Berlyand, M, Rybalko, Sandier 12)

AssumeΩsimply connected, and prescribe degree 1 on the boundary

ThenEεhas critical points forlargeε Remarks

∗ Probably holds also for degree≥2, but no proof

∗ We may work onΩ =D(price to pay: a weight in the potential)

∗ Rough plan: start fromε=∞, and perturb the problem

∗ Help from the minimum of the modulus principle

A tool: almost Blaschke products

Let

Ma= z−a

1−az, a∈D,z ∈D(Moebius transform) B_α,a1,...,ad =α

d

Y

j=1

M_aj, α∈S¹,a₁, . . . ,a_d ∈D(Blaschke product)

Blaschke products

Proposition

Blaschke products are precisely critical points ofE∞inDwith prescribed degreed >0

Hint

Compute the Hopf differential of critical points

Almost Blaschke products

Remarks

∗ Thus whenε=∞, critical points = energy minimizers (all Blaschke productsBhave energyE∞(B) =πd)

∗ Search of critical points whenε1 leads to maps of energy E_ε(u)−πd 1 (and thusE∞(u)−πd 1)

∗ Energetically, these are “almost” Blaschke products

∗ Their structure? Crucial matter for the existence of critical points

∗ Objects better fitted for analysis: traces onS¹, thus g:S¹→S¹s.t. degg=d and|g|²_H1/2−πd 1, where

|g|²_H1/2 := 1 2

Z

D

|∇u|², withuthe harmonic extension ofg

Structure of almost Moebius transforms

Theorem (Berlyand, M, Rybalko, Sandier 12)

Assume|g|²_H1/2 ≤π+δ <2π(i.e., no room for 2 Moebius transformsM_a) and degg =1

Then we may writeu=Mae^ıϕ with (1)|ϕ|_H1/2 ≤F(δ)

In addition, for smallδwe may pickas.t.g7→ais continuous Remark

The phase control part of the statement (estimate (1)) is not intuitively clear: if we takeg :S¹→S¹smooth and of zero degree, then its smooth phaseϕdoes not satisfy (1)

Structure of almost Moebius transforms

Sketch of proof for smallδ

∗ Minimum of the modulus principle=⇒there is no room for two zeros of the harmonic extensionuofg

∗ From this, control the region where|u| 6≈1, then the phase of u, then (by taking traces) the one ofg

∗ We may takea=the zero ofu. Continuity comes essentially from uniqueness

An application: killing the Moebius group

The set

X :={g∈H^1/2(S¹;S¹);degg =1}

is not weakly closed (troubles come from the action of the Moebius group)

However:

{g ∈X;|g|²_H1/2 ≤π+δ <2π}/Moebius group is weakly closed

Structure of almost Blaschke products

Theorem (Berlyand, M, Rybalko, Sandier 12) Letd ≥2

Then there existε,C>0 such that

g :S¹→S¹,|g|²_H1/2 ≤πd+ε, degu =d =⇒ u=B_1,a1,...,ade^ıϕ, with|ϕ|_H1/2 ≤C

Remark

Probably∀ε <2πworks...

Idea of proof

Induction, relying on the cased =1 + Wente estimates in order to obtain almost orthogonal decomposition of the energy

Back to the existence of critical points ofE_εin simply connected domains. Recall

Theorem (Berlyand, M, Rybalko, Sandier 12)

AssumeΩsimply connected, and prescribe degree 1 on the boundary

ThenE_εhas critical points forlargeε

Sketch of proof

∗ Min-max method: consider minF max

a∈D

{E_ε(F(a)); F ∈C(D;H¹),F(a) =Mafor|a| ≈1

|F(a)|=1 onS¹, ∀a∈D}

∗ Establish mountain pass geometry (relies on the structure of almost Moebius maps)

∗ Prove that the energy functional isC¹(small miracle)

∗ Next establish behavior of Palais-Smale (PS) sequences.

Requires killing the Moebius group (rescaling)

∗ Establish decomposition of the energy (bubbling). Relies on Wente estimates

∗ Identify all possible limits. Relies on the minimum of the modulus principle

∗ Establish compactness of PS sequences (and conclude)

All steps but the identification of the limit and can be performed for arbitraryε, degrees and multiply connected domains (via additional Wente type estimates)

This leads to bubbling analysis à la Brezis-Coron or Struwe, but not to compactness

A non local critical problem

How much it takes to wind once (or more) Find

mp=min{|g|^p

W^1/p,p;g :S¹→S¹,degg =1}

with

|g|^p

W^1/p,p = Z Z

S¹×S¹

|g(x)−g(y)|^p

|x −y|² dxdy

Main difficulty: non compact (energy invariant by the Moebius group)

Easy cases

p=1:m₁=2π, minimizers areW^1,1-maps with non decreasing phases

p=2:m₂=4π², minimizers are Moebius maps Proof whenp=2

Writeg =X

a_ne^ınθ. Then

|g|²_H1/2 =4π²X

|n||a_n|²

and

degg =X

n|a_n|²

Theorem (M 13)

There exists someε >0 such thatm_pis attained when p∈(2−ε,2)

Sketch of proof For suchp,

{g :S¹→S¹;degg =1,|g|^p

W^1/p,p ≈m_p}/Moebius group

is weakly closed

I do not know what happens when degg ≥2 (even forp close to 2)

Analysis for small ε

Theorem (Lamy, M 13)

LetΩbe simply connected, andd ≥1. Then, for smallε

∗ Under some (explicit) non degeneracy assumptions onΩ,E_ε has critical pointsu_εwith prescribed degreed

∗ In particular, critical points do exist whend =1 andΩis close to a disc

∗ Whend =1, the non degeneracy assumptions are

“generically” satisfied Remark

The non degeneracy assumptions look like generic ones. But we do not know whether they are indeeed generic whend ≥2 or in multiply connected domains

Strategy of the proof

∗ Assume existence of critical points. Find formal limit

∗ In the spirit of Bethuel, Brezis, Hélein 94, limit should be of the form

u₀(z) =

d

Y

j=1

z−a_j

|z−a_j|

e^ıH(z)

with unknowna₁, . . . ,a_d ∈ΩandH harmonic

∗ There is a formal relation betweena= (a₁, . . . ,a_d)and g:=tru₀: the configurationais a critical point of some appropriate renormalized energy (intuitively not so clear)

Strategy of the proof -ctd

∗ Next step consists in constructing critical points ofE_ε with Dirichlet boundary conditiong_ε≈gand “emanating from” a

“singular” configurationa_ε≈a

∗ This can be performed by either variational methods (Fang Hua Lin, Tai-Chia Lin 97, del Pino, Felmer 97) or gluing methods ( Pacard, Rivière 00 , del Pino, Kowalczyk, Musso 06)

∗ This requires a (first) nondegeneracy assumption

∗ Next findg_εs.t. the solution with boundary valueg_ε is a critical point with prescribed degrees

∗ This requires a (second) nondegeneracy assumption

∗ Existence ofgεis not obtained by inverse functions, but by Leray-Schauder degree theory

Strategy of the proof -ctd

∗ Up to now, everything adapts to arbitrary domains and degrees

∗ But we were able to prove genericity only whenΩis simply connected andd =1

∗ Even whend =1, nondegeneracy assumptions do not look generic if taken separately. But their couple is generically satisfied

∗ The last part relies on transversality results (à la Quinn 70)

Thank you for your attention!