Phase transition and symmetry breaking

Jean Dolbeault

http://www.ceremade.dauphine.fr/∼dolbeaul Ceremade, Universit´e Paris-Dauphine

November 6, 2019 FLACAM 2019

French Latin - American Conference on New Trends in Applied Mathematics

5 - 8 November 2019, Santiago, Chile

Outline

Phase transitionandsymmetry breaking, a warning Preliminaries: two observations

BPhase transition and asymptotic behaviour in aflockingmodel with a mean field term

BMoving planes and eigenvalues

Symmetry and symmetry breaking in interpolation inequalities BGagliardo-Nirenberg-Sobolev inequalities on the sphere B[Keller-Lieb-Thirring inequalities on the sphere]

BCaffarelli-Kohn-Nirenberg inequalities Ground states with magnetic fields

BMagnetic rings, a one-dimensional magnetic interpolation inequality BInterpolation inequalities in dimensions2and3, spectral estimates

Aharonov-Bohm magnetic fields inR²

Phase transition and symmetry breaking

The notion ofphase transition in physics

BEhrenfest’s classification and more recent definitions Symmetry breaking

BThe principles of Pierre Curie

BA mathematical point of view: the symmetry of the ground state Bifurcations, interpolation inequalities and evolution equations BSubcritical interpolation inequalities depending on a single parameter

BThe non-linear problemversusthe linearized spectral problem BNonlinear flows as a tool: generalized Bakry-Emery method BEnergy and relaxation

Preliminaries

Phase transition and asymptotic behaviour in aflockingmodel with a mean field term:

thehomogeneous Cucker-Smale / McKean-Vlasov model PhD thesis of Xingyu Li, https://arxiv.org/abs/1906.07517 Moving planes and eigenvalues

(Old) joint work with P. Felmer

A simple version of the Cucker-Smale model

A model for bird flocking (simplified version)

∂f

∂t =D∆vf +∇^v·(∇^vφ(v)f −uf f) whereu_f =´

v f dv is the average velocity f is a probability measure

left:φ(v) =1

4|v|⁴−1

2|v|² right:φ(v)−uf ·v

-2 -1 1 2

-1.0 -0.5 0.5 1.0 1.5 2.0

-2 -1 1 2

-2 2 4 6

[J. Tugaut, 2014]

[A. Barbaro, J. Ca˜nizo, J.A. Carrillo, and P. Degond, 2016]

Stationary solutions: phase transition in dimension d = 1

0.1 0.2 0.3 0.4 0.5 0.6

-1.0 -0.5 0.5 1.0

d = 1: there exists a bifurcation pointD=D_∗ such that the only stationary solution corresponds tou_f = 0 ifD>D_∗ and there are three solutions corresponding tou_f = 0,±u(D)ifD<D_∗

Phase transition in dimension d = 2

0.1 0.2 0.3 0.4

-1.0 -0.5 0.5 1.0

Dynamics and free energy

Thefree energy F[f] :=D

ˆ

R^d

f logf dv+ ˆ

R^d

fφdv−1 2|u_f|² decays according to

d

dtF[f(t,·)] =− ˆ

R^d

D∇^vf

f +∇^vφ−uf

2

f dv d = 1: ifF[f(t = 0,·)]<F[f?⁽⁰⁾] andD<D∗, then

F[f(t,·)]− Fh f_?^(±)i

≤C e^−λt

d = 1: λis the eigenvalue of the linearized problem atf_?^(±) in the weighted space L²

(f?^(±))⁻¹

with scalar product

High and low noise regimes

∂f

∂t =D∆f +∇ ·

(v−u_f)f +αv |v|²−1 f

Heret≥0 denotes the time variable,v∈R^d is the velocity variable u_f(t) =

´

R^dv f(t,v)dv

´

R^df(t,v)dv is themean velocity [J. Tugaut],[A. Barbaro, J. Canizo, J. Carrillo, P. Degond]

Theorem (X. Li)

Let d≥1andα >0. There exists a critical D∗>0 such that (i) D>D_∗: only one stable stationary distribution withu_f =0 (ii) D<D_∗: one instable isotropic stationary distribution withu_f =0 and a continuum of stable non-negative non-symmetric polarized stationary distributions (unique up to a rotation)

Relative entropy and related quantities

Free energy F[f] :=D

ˆ

R^d

f logf dv+ ˆ

R^d

fφαdv−1 2|uf|² Relative entropy with respect to a stationary solutionfu

F[f]− F[fu] =D ˆ

R^d

f log f

f_u

dv−1

2|u_f −u|² Relative Fisher information

I[f] :=

ˆ

R^d

D ∇f

f +αv|v|²+ (1−α)v−u_f

2

f dv Thelocal Gibbs state

−¹(^{1|v−u|2+α|v|4−α|v|2})

Gibbs state versus stationary solution

F[f]is a Lyapunov function in the sense that d

dtF[f(t,·)] =− I[f(t,·)]

whereF[f]− F[fu] =D ˆ

R^d

f log f

fu

dv−1

2|u_f −u|²and

I[f] =D² ˆ

R^d

∇log

f G_f

2

f dv

d

dtF[f(t,·)] = 0if and only iff =Gf is a stationary solution

Stability and coercivity

Q_1,u[g] := lim

ε→0

2 ε²F

fu(1 +εg)

=D ˆ

R^d

g²fudv−D²|vg|² wherevg := _D¹ ´

R^dv g fudv Q_2,u[g] := lim

ε→0

1 ε²I

fu(1 +εg)

=D² ˆ

R^d

|∇g−vg|²fudv Stability: Q_1,u≥0 ?

Coercivity: Q_2,u≥λQ_1,u for someλ >0 ?

Stability of the isotropic stationary solution

Q_1,0[g] =D ˆ

R^d

g²f₀dv−D²|v_g|²

We consider the space of the functionsg ∈L²(f0dv)such that ˆ

R^d

g f0dv = 0

Lemma (X. Li)

Q1,0is a nonnegative quadratic form if and only if D≥D∗ and Q_1,0[g]≥η(D)

ˆ

R^d

g²f0dv for some explicitη(D)>0 if D>D_∗

Stability of the polarized stationary solution

Corollary (X. Li)

F has a unique nonnegative minimizer with unit mass, f₀, if D ≥D_∗. Otherwise, if D<D_∗, we have

minF[f] =F[fu]<F[f0] for any u∈R^d such that|u|=u(D).

The minimum is taken on L¹₊ R^d,(1 +|v|⁴)dv

such that´

R^df dv= 1 Corollary (X. Li)

Let D<D_∗,|u|=u(D)6= 0. Then

A coercivity result

Poincar´e inequality: if´

R^dh fudv= 0 ˆ

R^d

|∇h|²fudv≥ΛD

ˆ

R^d

|h|²fudv Letf ∈L¹(R^d)with´

R^df dv = 1,g = (f −fu)/fu and let u[f] = ^u(D)_|uf| uf ifD <D∗ anduf 6=0. Otherwise takeu[f] =0 Proposition (X. Li)

Let d≥1,α >0, D >0. Ifu=0, then Q_2,u[g]≥ C^DQ_1,u[g]

Otherwise, if|u|=u(D)6= 0 for some D∈(0,D∗), then Q_2,u[g]≥ C^D 1−κ(D) (vg·u)²

|v_g|²|u|²Q_1,u[g]

High noise: convergence to the isotropic solution

Theorem (X. Li)

For any d≥1and anyα >0, if D >D_∗, then for any solution f with nonnegative initial datum fin of mass1 such thatF[fin]<∞, there is a positive constant C such that, for any time t>0,

0≤ F[f(t,·)]− F[f0]≤C e^−CDt

Non-local scalar product and linearized evolution operator

In terms off =f0(1 +g)the evolution equation is f0

∂g

∂t =D∇ ·

(∇g−vg)f0−vgg f0

withvg =_D¹´

R^dv g f0dv,Q_1,0[g] =hg,gi=D´

R^dg g f0dv−D²vg·vg

h·,·iis a scalar productonX :=

g ∈L²(f₀dv) : ´

R^dg f₀dv= 0

∂g

∂t =Lg−vg·

D∇g−(v+∇φα)g

, Lg :=D∆g−(v+∇φα)·(∇g−vg)

Lemma (X. Li)

Assume that D>D_∗ andα >0. The norm g7→p

hg,giis equivalent to the standard norm onL²(f0dv)

The linearized operatorLis self-adjoint onX and

− hg,Lgi=Q2,0[g]

The scalar producth·,·iis well adapted to the linearized evolution operator in the sense that a solution of thelinearized equation

∂g

∂t =Lg with initial datumg0∈ X is such that

1 2

d

dtQ1,0[g] = 1 2

d

dthg,gi=hg,Lgi=−Q2,0[g] and has exponential decay. We know that

hg(t,·),g(t,·)i=hg0,g0ie^−2CDt ∀t ≥0

Low noise and partially symmetric solutions

Proposition (X. Li)

Letα >0, D>0and consider a solution f ∈C⁰ R⁺,L¹(R^d) with initial datum fin∈L¹₊(R^d)such that F[fin]<F[f0] and

uf_in = (u,0. . .0)for some u6= 0. We further assume that

fin(v1,v2, . . .vi−1,vi, . . .) =fin(v1,v2, . . .vi−1,−vi, . . .)for any i = 2, 3,. . . d . Then

lim sup

t→+∞

e^+λtkf(t,·)−fukL¹(R^d)<∞ 0≤ F[f(t,·)]− F[fu]≤C e^−λt ∀t ≥0 holds withλ=C^D 1−κ(D)

>0

Without symmetry assumption, the question of the rate of convergence to a solution / to the set of polarized solutions is still open

Moving planes and eigenvalues

(Old) joint work with P. Felmer

The theorem of Gidas, Ni and Nirenberg

Theorem

[Gidas, Ni and Nirenberg, 1979 and 1980]Let u∈C²(B), B=B(0,1)⊂R^d, be a solution of

∆u+f(u) = 0in B, u= 0on∂B

and assume that f isLipschitz. If u ispositive, then it isradially symmetric and decreasingalong any radius: u⁰(r)<0for any r ∈(0,1]

Extension: ∆u+f(r,u) = 0,r =|x|if ^∂f_∂r ≤0... a “cooperative” case

An extension of the theorem of Gidas, Ni and Nirenberg

Theorem (JD, P. Felmer, 1999)

∆u+λf(r,u) = 0in B, u= 0on∂B

and assume that f ∈C¹(R⁺×R⁺)(no assumption on the sign of ^∂f_∂r) There existsλ1,λ2with0< λ1≤λ2such that

(i)Monotonicity: ifλ∈(0, λ1), then _dr^d(u−λu0)<0where u0is the solution of∆u0+λf(r,0) = 0

(ii)Symmetry: ifλ∈(0, λ2), then u is radially symmetric

Symmetry and symmetry breaking in interpolation inequalities

without magnetic field

Gagliardo-Nirenberg-Sobolev inequalities on the sphere [Keller-Lieb-Thirring inequalities on the sphere]

Caffarelli-Kohn-Nirenberg inequalities

Joint work with M.J. Esteban, M. Loss, M. Kowalczyk,...

A result of uniqueness on a classical example

On the sphereS^d, let us consider the positive solutions of

−∆u+λu=u^p−1 p∈[1,2)∪(2,2^∗]ifd≥3,2^∗=_d−2^2d

p∈[1,2)∪(2,+∞)ifd = 1,2 Theorem

Ifλ≤d , u≡λ^1/(p−2)is the unique solution

[Gidas & Spruck, 1981],[Bidaut-V´eron & V´eron, 1991]

Bifurcation point of view and symmetry breaking

2 4 6 8 10

2 4 6 8

Figure:(p−2)λ7→(p−2)µ(λ)with d= 3

k∇uk²L²(S^d)+λkuk²L²(S^d)≥µ(λ)kuk²L^p(S^d)

Taylor expansion ofu= 1 +ε ϕ1as ε→0with−∆ϕ1=dϕ1

µ(λ)< λ if and only if λ > _p−2^d

BThe inequality holds withµ(λ) =λ= _p−2^d [Bakry & Emery, 1985]

[Beckner, 1993],[Bidaut-V´eron & V´eron, 1991, Corollary 6.1]

The Bakry-Emery method on the sphere

Entropy functional E^p[ρ] := _p−2¹ h´

S^dρ^2p dµ− ´

S^dρdµ²_pi

if p6= 2 E²[ρ] :=´

S^dρlog

ρ kρk_{L1 (Sd)}

dµ Fisher informationfunctional

I^p[ρ] :=´

S^d|∇ρ^1p|²dµ

[Bakry & Emery, 1985]carr´e du champmethod: use the heat flow

∂ρ

∂t = ∆ρ and observe that _dt^dE^p[ρ] =− I^p[ρ]

d

The evolution under the fast diffusion flow

To overcome the limitationp≤2^#, one can consider a nonlinear diffusion of fast diffusion / porous medium type

∂ρ

∂t = ∆ρ^m

[Demange],[JD, Esteban, Kowalczyk, Loss]: for anyp∈[1,2^∗] K^p[ρ] := d

dt

I^p[ρ]−dE^p[ρ]

≤0

1.0 1.5 2.5 3.0

0.0 0.5 1.5 2.0

(p,m)admissible region,d = 5

References

JD, M. J. Esteban, M. Kowalczyk, and M. Loss. Improved interpolation inequalities on the sphere. Discrete and Continuous Dynamical Systems Series S, 7 (4): 695-724, 2014.

JD, M.J. Esteban, and M. Loss. Interpolation inequalities on the sphere: linearversusnonlinear flows. Annales de la facult´e des sciences de Toulouse S´er. 6, 26 (2): 351-379, 2017

JD, M.J. Esteban. Improved interpolation inequalities and stability. Preprint, 2019 arXiv:1908.08235

Optimal inequalities

Withµ(λ) =λ= _p−2^d : [Bakry & Emery, 1985]

[Beckner, 1993],[Bidaut-V´eron & V´eron, 1991, Corollary 6.1]

k∇uk²L²(S^d)≥ d p−2

kuk²L^p(S^d)− kuk²L²(S^d)

∀u∈H¹(S^d)

d ≥3,p∈[1,2)orp∈(2,_d−2^2d ) d = 1ord= 2,p∈[1,2)orp∈(2,∞)

p=−2 = 2d/(d−2) =−2withd= 1 [Exner, Harrell, Loss, 1998]

k∇uk²L²(S¹)+1 4

ˆ

S¹

1 u² dµ

−1

≥ 1

4kuk²L²(S¹) ∀u∈H¹+(S¹)

Caffarelli-Kohn-Nirenberg, symmetry and symmetry breaking results, and weighted nonlinear flows

Joint work with M.J. Esteban and M. Loss

Critical Caffarelli-Kohn-Nirenberg inequality

LetD^a,b:=n

v ∈L^p R^d,|x|^−bdx

: |x|^−a|∇v| ∈L² R^d,dxo ˆ

R^d

|v|^p

|x|^{b p} dx 2/p

≤ Ca,b

ˆ

R^d

|∇v|²

|x|^2a dx ∀v ∈ D^a,b holds under conditions ona andb

p= 2d

d−2 + 2 (b−a) (critical case) BAn optimal function among radial functions:

v?(x) =

1 +|x|^{(p−2) (ac−a)}−_p−2²

and C^?_a,b= k |x|^−bv_?k²p

k |x|^−a∇v_?k²2

Question: Ca,b= C^?_a,b (symmetry) orCa,b>C^?_a,b (symmetry breaking) ?

Critical CKN: range of the parameters

Figure: d = 3 ˆ

R^d

|v|^p

|x|^{b p} dx 2/p

≤ Ca,b

ˆ

R^d

|∇v|²

|x|^2a dx

a b

0 1

−1

b=a

b=a+ 1

a=^d−2₂ p

p= 2d

d−2 + 2 (b−a) a<a_c := (d−2)/2

a≤b≤a+ 1ifd≥3, [Il’in (1961)]

[Glaser, Martin, Grosse, Thirring (1976)]

Linear instability of radial minimizers:

the Felli-Schneider curve

The Felli & Schneider curve b_FS(a) := d(ac−a)

2p

(ac−a)²+d−1 +a−a_c

a b

0

[Smets],[Smets, Willem],[Catrina, Wang],[Felli, Schneider]

v 7→C^?_a,b ˆ

R^d

|∇v|²

|x|^2a dx− ˆ

R^d

|v|^p

|x|^{b p} dx 2/p

is linearly instable atv =v_?

Symmetry versus symmetry breaking:

the sharp result in the critical case

[JD, Esteban, Loss (2016)]

a b

0

Theorem

Let d≥2and p<2^∗. If either a∈[0,ac)and b>0, or a<0 and

The symmetry proof in one slide

Achange of variables: v(|x|^α−1x) =w(x),Dαv= α^∂v_∂s,¹_s∇^ωv kvk^L2p,d−n(R^d) ≤Kα,n,pkDαvk^ϑL^2,d−n(R^d)kvk^1−ϑL^p+1,d−n(R^d) ∀v ∈H^p_d−n,d−n(R^d)

Concavity of the R´enyi entropy power: with L^α=−D^∗_αDα=α² u⁰⁰+ⁿ⁻¹_s u⁰

+_s¹2∆ωuand ^∂u_∂t =L^αu^m

−dt^d G[u(t,·)] ´

R^du^mdµ1−σ

≥(1−m) (σ−1)´

R^du^m L^αP−

´

Rd´u|DαP|²dµ

Rdu^mdµ

2

dµ + 2´

R^d

α⁴ 1−¹n

P⁰⁰−^Ps⁰ −α²(n−1)^∆ωPs²

2

+^2α_s2²

∇^ωP⁰−^∇ωs^P

2 u^mdµ + 2´

R^d

(n−2) α²_FS−α²

|∇^ωP|²+c(n,m,d)^|∇_P^ω2^P|4

u^mdµ Elliptic regularity and the Emden-Fowler transformation: justifying theintegrations by parts

The variational problem on the cylinder

BWith the Emden-Fowler transformation

v(r, ω) =r^a−acϕ(s, ω) with r =|x|, s=−logr and ω= x r the variational problem becomes

Λ7→µ(Λ) := min

ϕ∈H¹(C)

k∂sϕk²L²(C)+k∇^ωϕk²L²(C)+ Λkϕk²L²(C)

kϕk²L^p(C)

is a concave increasing function Restricted to symmetric functions, the variational problem becomes

µ?(Λ) := min

ϕ∈H¹( )

k∂sϕk²L²(R^d)+ Λkϕk²L²(R^d)

kϕk² =µ?(1) Λ^α

Numerical results

20 40 60 80 100

10 20 30 40 50

symmetric non-symmetric asymptotic

bifurcation µ

Λ^α µ(Λ)

(Λ) =µ(1) Λ^α

Parametric plot of the branch of optimal functions forp= 2.8, d= 5.

Non-symmetric solutions bifurcate from symmetric ones at a bifurcation pointΛ1computed by V. Felli and M. Schneider. The branch behaves for large values ofΛ as shown by F. Catrina and Z.-Q. Wang

Three references

Lecture notes onSymmetry and nonlinear diffusion flows...

a course on entropy methods (see webpage)

[JD, Maria J. Esteban, and Michael Loss]Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs

... the elliptic point of view: Proc. Int. Cong. of Math., Rio de Janeiro, 3: 2279-2304, 2018.

[JD, Maria J. Esteban, and Michael Loss]Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization... the parabolic point of view

Journal of elliptic and parabolic equations, 2: 267-295, 2016.

With magnetic fields (1/3) in dimensions 2 and 3

Interpolation inequalities and spectral estimates

Estimates, numerics; an open question on constant magnetic fields

Magnetic interpolation inequalities in the Euclidean space

BThree interpolation inequalities and their dual forms BEstimates in dimensiond = 2for constant magnetic fields

Lower estimates

Upper estimates and numerical results

A linear stability result (numerical) and an open question

Assumptions are not detailed: A∈L^d+ε_loc (R^d),ε >0+ integral conditions as in[Esteban, Lions, 1989]

Estimates are given (almost) only in the casep>2 but similar estimates hold in the other cases

Magnetic Laplacian and spectral gap

In dimensionsd= 2andd = 3: themagnetic Laplacianis

−∆Aψ=−∆ψ−2iA· ∇ψ+|A|²ψ−i(divA)ψ

where the magnetic potential (resp. field) isA(resp.B=curlA) and H¹A(R^d) :=

ψ∈L²(R^d) : ∇^Aψ∈L²(R^d) , ∇^A:=∇+iA Spectral gap inequality

k∇^Aψk²L²(R^d)≥Λ[B]kψk²L²(R^d) ∀ψ∈H¹A(R^d) Λdepends only on B=curlA

Assumption: equality holds for someψ∈H¹_A(R^d) IfBis a constant magnetic field,Λ[B] =|B|

Ifd= 2, spec(−∆A) ={(2j+ 1)|B| : j∈N} is generated by the Landau levels. TheLowest Landau Levelcorresponds to j= 0

Magnetic interpolation inequalities

k∇^Aψk²L²(R^d)+αkψk²L²(R^d)≥µB(α)kψk²L^p(R^d) ∀ψ∈H¹_A(R^d) for anyα∈(−Λ[B],+∞)and anyp∈(2,2^∗),

k∇^Aψk²L²(R^d)+βkψk²L^p(R^d)≥νB(β)kψk²L²(R^d) ∀ψ∈H¹_A(R^d) for anyβ∈(0,+∞)and any p∈(1,2)

k∇^Aψk²L²(R^d)≥γ ˆ

R^d

|ψ|²log |ψ|² kψk²L²(R^d)

!

dx+ξB(γ)kψk²L²(R^d)

(limit case corresponding top= 2) for anyγ∈(0,+∞)

Cp:=



 minu∈H¹(R^d)\{0}

k∇uk²_{L2 (Rd}

)+kuk²_{L2 (Rd}

)

kuk²

Lp(Rd)

if p∈(2,2^∗)

min ^k∇uk

2

L2 (Rd)+kuk²

Lp(Rd) if ∈(1,2)

A statement

Theorem

p∈(2,2^∗): µBis monotone increasing on(−Λ[B],+∞), concave and

α→(−Λ[B])lim +

µB(α) = 0 and lim

α→+∞µB(α)α^{d−22 −dp} = Cp

p∈(1,2): νB is monotone increasing on(0,+∞), concave and

β→0lim₊νB(β) = Λ[B] and lim

β→+∞νB(β)β^{−2p+d2(2−p)p} = Cp

ξBis continuous on(0,+∞), concave, ξB(0) = Λ[B] and ξB(γ) =^d₂γ log ^π_γ^e2

(1 +o(1)) as γ→+∞ Constant magnetic fields: equality is achieved

Nonconstant magnetic fields: only partial answers are known

2 4 6 8 10 2

4 6 8

Numerical results

and the symmetry issue

2 4 6 8 10

-1 1 2 3

0.004 0.003 0.002 0.001 0.001 0.002 0.003

-0.02

-0.04

-0.06

-0.08

2 4

-1

6

Figure: Cased= 2,p= 3,B= 1

An open question of symmetry

[Bonheure, Nys, Van Schaftingen, 2016]for a fixed α >0 and forBsmall enough, the optimal functions are radially symmetric functions,i.e., belong toC⁰

This regime is equivalent to the regime asα→+∞for a givenB, at least if the magnetic field is constant

Numerically our upper and lower bounds are (in dimensiond = 2, for a constant magnetic field) numerically extremely close

The optimal function in C⁰is linearly stable with respect to perturbations inC¹

A reference: JD, M.J. Esteban, A. Laptev, M. Loss. Interpolation inequalities and spectral estimates for magnetic operators. Annales Henri Poincar´e, 19 (5): 1439-1463, May 2018

BProve that the optimality case is achieved among radial function if d= 2andB is a constant magnetic field

With magnetic fields (2/3) Magnetic rings: the case of S ¹

BA magnetic interpolation inequality onS¹: withp>2 kψ⁰+i aψk²L²(S¹)+αkψk²L²(S¹) ≥µa,p(α)kψk²L^p(S¹)

BConsequences

[A Keller-Lieb-Thirring inequality]

A new Hardy inequality for Aharonov-Bohm magnetic fields inR²

Magnetic flux, a reduction

Assume thata:R→Ris a2π-periodic function such that its restriction to(−π, π]≈S¹is in L¹(S¹)and define the space

Xa :=

ψ∈Cper(R) : ψ⁰+i aψ∈L²(S¹)

A standard change of gauge (see e.g.[Ilyin, Laptev, Loss, Zelik, 2016])

ψ(s)7→eⁱ

´s

−π(a(s)−¯a)dσ

ψ(s) wherea¯:=´π

−πa(s)dσis themagnetic flux, reduces the problem to ais a constant function

For anyk ∈Z,ψbys7→e^iksψ(s)shows thatµa,p(α) =µk+a,p(α) a∈[0,1]

µa,p(α) =µ1−a,p(α)because

|ψ⁰+i aψ|²=|χ⁰+i(1−a)χ|²=

ψ⁰−i aψ

2ifχ(s) =e^−isψ(s) a∈[0,1/2]

Optimal interpolation

We want to characterize theoptimal constant in the inequality kψ⁰+i aψk²L²(S¹)+αkψk²L²(S¹) ≥µa,p(α)kψk²L^p(S¹)

written for anyp>2,a∈(0,1/2],α∈(−a²,+∞),ψ∈Xa

µa,p(α) := inf

ψ∈Xa\{0}

´π

−π |ψ⁰+i aψ|²+α|ψ|² dσ kψk²L^p(S¹)

p=−2 = 2d/(d−2)withd = 1[Exner, Harrell, Loss, 1998]

p= +∞[Galunov, Olienik, 1995] [Ilyin, Laptev, Loss, Zelik, 2016]

limα→−a²µa,p(α) = 0[JD, Esteban, Laptev, Loss, 2016]

Using a Fourier seriesψ(s) =P

k∈Zψke^iks, we obtain that

0 X

An interpolation result for the magnetic ring

Theorem

For any p>2, a∈R, andα >−a²,µa,p(α)is achieved and

(i) if a∈[0,1/2]and a²(p+ 2) +α(p−2)≤1, thenµa,p(α) =a²+α and equality is achieved only by the constant functions

(ii) if a∈[0,1/2]and a²(p+ 2) +α(p−2)>1, thenµa,p(α)<a²+α and equality is not achieved among the constant functions

Ifα >−a², a7→µa,p(α)is monotone increasing on(0,1/2)

-0.2 -0.1 0.1 0.2 0.3 0.4

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.2 0.4 0.6 0.8 1.0 1.2

Figure:α7→µa,p(α) withp= 4 and (left)a= 0.45 or (right)a= 0.2

Elimination of the phase

Let us define

Q^a,p,α[u] := ku⁰k²L²(S¹)+a²ku⁻¹k⁻²L²(S¹)+αkuk²L²(S¹)

kuk²L^p(S¹)

Lemma

For any a∈(0,1/2), p>2,α >−a², µa,p(α) = min

u∈H¹(S¹)\{0}Q^a,p,α[u]

is achieved by a function u>0

A new Hardy inequality

ˆ

R²

|(i∇+a) Ψ|²dx≥τ ˆ

R²

ϕ(x/|x|)

|x|² |Ψ|²dx ∀ϕ∈L^q(S¹), q∈(1,+∞) Corollary

Let p>2, a∈[0,1/2], q=p/(p−2)and assume thatϕis a

non-negative function inL^q(S¹). Then the inequality holds with τ >0 given by

αa,p τkϕk^Lq(S¹)

= 0

Moreover,τ=a²/kϕk^Lq(S¹) if4a²+kϕk^Lq(S¹)(p−2)≤1

For anya∈(0,1/2), by takingϕconstant, small enough in order that 4a²+kϕk^Lq(S¹)(p−2)≤1, we recover the inequality

ˆ

R²

|(i∇+a) Ψ|²dx≥a² ˆ

R²

|Ψ|²

|x|² dx

[Laptev, Weidl, 1999]constant magnetic fields;[Hoffmann-Ostenhof, Laptev, 2015]in R^d,d≥3

With magnetic fields (3/3) Aharonov-Bohm magnetic fields

in R ²

Aharonov-Bohm effect

[Interpolation and Keller-Lieb-Thirring inequalities inR²] Aharonov-Symmetry and symmetry breaking

Aharonov-Bohm effect

A major difference between classical mechanics and quantum mechanics is that particles are described by a non-local object, the wave function. In 1959 Y. Aharonov and D. Bohm proposed a series of experiments intended to put in evidence such phenomena which are nowadays calledAharonov-Bohm effects

One of the proposed experiments relies on a long, thin solenoid which produces a magnetic field such that the region in which the magnetic field is non-zero can be approximated by a line in dimensiond= 3 and by a point in dimensiond= 2

B[Physics today, 2009]“The notion, introduced 50 years ago, that electrons could be affected by electromagnetic potentials without coming in contact with actual force fields was received with a skepticism that has spawned a flourishing of experimental tests and expansions of the original idea.” Problem solved by considering appropriate weak solutions !

BIs the wave function a physical object or is its modulus ? Decisive experiments have been done only 20 years ago

The interpolation inequality

Let us consider an Aharonov-Bohm vector potential A(x) = a

|x|²(x2,−x1), x= (x1,x2)∈R²\ {0}, a∈R Magnetic Hardy inequality[Laptev, Weidl, 1999]

ˆ

R²

|∇^Aψ|²dx≥min

k∈Z(a−k)² ˆ

R²

|ψ|²

|x|²dx where∇^Aψ:=∇ψ+ iAψ, so that, withψ=|ψ|e^iS

ˆ

R²

|∇^Aψ|²dx= ˆ

R²

(∂r|ψ|)²+ (∂rS)²|ψ|²+ 1

r²(∂θS+A)²|ψ|²

dx Magnetic interpolation inequality

ˆ

|∇ |²

ˆ |ψ|²

≥

ˆ |ψ|^p 2/p

A magnetic Hardy-Sobolev inequality

Theorem

Let a∈[0,1/2]and p>2. For anyλ >−a², there is an optimal, monotone increasing, concave functionλ7→µ(λ)which is such that

ˆ

R²

|∇^Aψ|²dx+λ ˆ

R²

|ψ|²

|x|² dx≥µ(λ) ˆ

R²

|ψ|^p

|x|² dx 2/p

Ifλ≤λ?= 4¹⁻⁴_p2−4^a2 −a²equality is achieved by

ψ(x) = (|x|^α+|x|^−α)^−p−22 ∀x ∈R², with α=^p−2₂ √ λ+a² Ifλ > λ•with

λ•:= ⁸

√_p4

−a²(p−2)²(p+2) (3p−2)+2

−4p(p+4)

(p−2)³(p+2) −a²

there is symmetry breaking: optimal functions are not radially symmetric

0.1 0.2 0.3 0.4 0.5

-0.2 -0.1 0.1 0.2 0.3

Figure: Casep= 4

Symmetry breaking region:λ > λ•(a) Symmetry breaking region:λ < λ?

0.001 0.002 0.003 0.004 0.005

References

D. Bonheure, J. Dolbeault, M.J. Esteban, A. Laptev, M. Loss.

Inequalities involving Aharonov-Bohm magnetic potentials in dimensions 2 and 3. Preprint arXiv:1902.06454

D. Bonheure, J. Dolbeault, M.J. Esteban, A. Laptev, M. Loss.

Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields. Communications in Mathematical Physics, (2019).

J. Dolbeault, M. J. Esteban, A. Laptev, and M. Loss. Magnetic rings. Journal of Mathematical Physics, 59 (5): 051504, 2018.

J. Dolbeault, M. J. Esteban, A. Laptev, and M. Loss. Interpolation inequalities and spectral estimates for magnetic operators. Annales Henri Poincar, 19 (5): 1439-1463, May 2018.

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