Minimizers of non local energies, and ellipses Joan Verdera - - PowerPoint PPT Presentation

Minimizers of non local energies, and ellipses

Joan Verdera Universitat Aut`

• noma de Barcelona

Garnett-Marshall Conference, August 19-23 , 2019

Joan Verdera

• Minimizers. . .

The ellipse law : Kirchhoff meets dislocations

Coauthors : Carrillo (London), Mateu (Barcelona), Mora (Pavia), Rondi(Milano), Scardia (Edinburgh)

Joan Verdera

• Minimizers. . .

The Euler equation in the plane

v(z, t) velocity field of an ideal incompressible fluid (E)          ∂tv(z, t) + (v · ∇)v(z, t) = −∇p(z, t) div v = 0 v(z, 0) = v0(z) v · ∇ = v1∂1 + v2∂2

Joan Verdera

• Minimizers. . .

Acceleration and force

Particle trajectory : dz(t)/dt = v(z(t), t) d2z dt2 = ∂tv(z(t), t) + ∂1 v(z(t), t) v1(z(t), t) + . . . = ∂tv(z, t) + (v · ∇)v(z, t) Pressure and force force on blob V =

• ∂V

−p n dS =

• V

−∇p dx

Joan Verdera

• Minimizers. . .

Incompressibility : the velocity field is divergence free 0 =

• ∂V

v · n dS =

• V

div(v) dx

Joan Verdera

• Minimizers. . .

Vorticity

ω = curl(v) = ∂1v2 − ∂2v1 2 ∂v = 2 ∂ ∂z v = div v + i curl v = iω 1 πz is the fundamental solution of ∂ = ∂ ∂z v(z, t) = i 2 1 π¯ z ∗ ω = i 2π ω(ζ, t) ¯ z − ¯ ζ dA(ζ)

Joan Verdera

• Minimizers. . .

The vorticity equation

           ∂tω + (v · ∇)ω = 0 v = i 2π 1 ¯ z ∗ ω = ∇⊥N ∗ ω, N =

1 2π log |z|

ω(z, 0) = ω0(z) Particle trajectory : dz(t)/dt = v(z(t), t) dω(z(t), t) dt = ∂tω(z(t), t) + ∂1ω(z(t), t)v1(z(t), t) + . . .

Joan Verdera

• Minimizers. . .

Yudovich’s Theorem The vorticity equation is well posed in L∞ : For each ω0 ∈ L∞

c (C)

there is a unique global ”weak” solution to the vorticity equation with initial condition ω0. “Proof” : Solve dX(z, t) dt = v(X(z, t), t), X(z, 0) = z and set ω(z, t) = ω0(X−1(z, t))

• r

ω(X(z, t), t) = ω0(z)

Joan Verdera

• Minimizers. . .

Vortex patches

ω0 = χD, D a domain ω(z, t) = χD(t)(z)

D(t) D

Joan Verdera

• Minimizers. . .

The two known explicit examples

If D = D(0, 1) is the unit disc, then Dt = D(0, 1), 0 < t, χD(0,1)(z) is a steady solution to the vorticity equation If D0 = {(x, y) : x2/a2 + y2/b2 < 1} is an ellipse then Kirchhoff: Dt = eiΩt D0, 0 < t, Ω = ab (a + b)2 (vortex)

Joan Verdera

• Minimizers. . .

Why ellipses rotate ?

If E = {(x, y) : x2/a2 + y2/b2 ≤ 1} is an ellipse then 1 πz ⋆ χE

• (z) = ¯

z − λz, z ∈ E λ = a − b a + b

Joan Verdera

• Minimizers. . .

The equation of V-states

A domain D rotates with angular velocity Ω if and only if 2Ωz − 1 π¯ z ⋆ χD

• (z),

τ(z) = 0, z ∈ ∂D If D is an ellipse with semiaxis a and b the equation is 2Ωz − (z − λ¯ z), τ(z) = 0, z ∈ ∂D

Joan Verdera

• Minimizers. . .

Rotating vortex patches or V-states

Definition A V-state is a vortex patch that rotates with constant angular

• velocity. If the initial domain D0 has the origin as center of mass

Dt = eiΩ tD0 for a certain angular velocity Ω Deem–Zabuski (1978): numerical discovery of existence of V-states with m-fold symmetry Burbea (1981): analytical proof Hmidi-Mateu-V (2015): A V-state ”close” to the disc has boundary of class C∞ (and it is convex)

Joan Verdera

• Minimizers. . .

Joan Verdera

• Minimizers. . .

Classical Potential Theory

The Coulomb force field of a charge distribution µ in R3

• F(x) = −∇Pµ(x) =
• R3

x − y |x − y|3 dµ(y) The potential energy of a proof charge at x due to µ is Pµ(x) =

• R3

1 |x − y| dµ(y) The energy of a charge distribution µ E(µ) =

• R3×R3

1 |x − y| dµ(x) dµ(y)

Joan Verdera

• Minimizers. . .

Classical Potential Theory in the plane

• F(z) = −∇Pµ(z) =
• C

z − w |z − w|2 dµ(w) Pµ(z) = −

• C

log |z − w| dµ(w) E(µ) = −

• C×C

log |z − w| dµ(z) dµ(w) Books by Carleson and Wermer

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• Minimizers. . .

Potential Theory in the plane with an external field

One considers a confinement force (book by Saff and Totik)

• F(z) = −∇Pµ(z) = −2¯

∂Pµ(z) =

• C

z − w |z − w|2 dµ(w) − z Pµ(z) =

• C

− log |z − w| dµ(w) + |z|2 2 +

• C

|w|2 2 dµ(w) I(µ) = −

• C×C

log |z − w| dµ(z) dµ(w) +

• |z|2 dµ(z)

Joan Verdera

• Minimizers. . .

Potential Theory in the plane with an external field

Frostman There exists a unique probability measure that minimises the energy I(µ), namely, the normalised characteristic function of the unit disc D :

1 πχD(z)

The minimiser is unique. Then you proceed by a direct calculation

• r by symmetry plus harmonicity

Joan Verdera

• Minimizers. . .

IDEALISED DISLOCATIONS Straight & parallel; edge Single slip, and single sign (all positive) INTERACTION STRESS generated by a dislocation at 0 and acting on a dislocation at z F(z) = −κ ∇V (z) V (z) = − log |z| + x2

|z|2

x y

Conjecture: positive dislocations prefer to form vertical walls. Can we prove it ?

Joan Verdera

• Minimizers. . .

Energy of a dislocation : anisotropy

z = x + iy w = u + iv

• F(z) = −∇Pµ(z) = ∇
• C

(− log |z − w| + (x − u)2 |z − w|2 ) dµ(w)

• − z

Pµ(z)=

• C
• − log |z − w|+ (x − u)2

|z − w|2

• dµ(w)+|z|2

2 +

• C

|w|2 2 dµ(w) I(µ)=

• C×C
• − log |z − w|+ (x − u)2

|z − w|2

• dµ(z)dµ(w)+
• |z|2 dµ(z)

Joan Verdera

• Minimizers. . .

• M. G. Mora, L. Rondi, L. Scardia (2016)

It was a long standing conjecture that the unique minimiser of

I(µ) = −

• C×C
• − log |z − w| + (x − u)2

|z − w|2

• dµ(z) dµ(w)+
• |z|2 dµ(z)

was the semicircle law on the vertical axis µ := δ0 ⊗ 1 π

• 2 − y2 χ[−

√ 2, √ 2](y) dy

Joan Verdera

• Minimizers. . .

The problem

Take the interaction potential given by convolution with the kernel W(z) = Wα(z) = − log |z| + α x2 |z|2 , −1 ≤ α ≤ 1 coupled with the confinement term |z|2

Iα(µ)=−

• C×C
• − log |z − w|+α(x − u)2

|z − w|2

• dµ(z) dµ(w)+
• |z|2 dµ(z)

Joan Verdera

• Minimizers. . .

Carrillo, Mateu, Mora, Rondi, Scardia, V

The unique minimiser of the energy Iα is the normalized characteristic function of the ellipse centered at the origin with semi-axis √ 1 − α and √ 1 + α

• 1.5
• 1
• 0.5

0.5 1 1.5

• 1
• 0.5

0.5 1

Joan Verdera

• Minimizers. . .

Euler conditions

(Wα ∗ µ)(z) =

• C
• − log |z − w| + α(x − u)2

|z − w|2

• dµ(w)

Pµ(z) = (Wα ∗ µ)(z) + |z|2 2 + 1 2

• C

|w|2 dµ(w) (EL1) Pµ(z) = Cα, Cap a.e. on the support of µ (EL2) Pµ(z) ≥ Cα,

• ff support of µ

Joan Verdera

• Minimizers. . .

Sufficiency of Euler conditions

The functional I(µ) is strictly convex, because of the positivity of the Fourier transform of Wα on test functions ϕ with 0 integral

• (Wα)(ξ)ϕ(ξ) dA(ξ) = 1

2π (1 − α)ξ2

1 + (1 + α)ξ2 2

|ξ|4 ϕ(ξ) dA(ξ)

• (Wα ∗ ν) dν =
• (Wα)(ξ)|

ν(ξ)|2 dξ, ν(1) = 0 Strict convexity implies uniqueness of the minimiser and sufficiency

• f Euler’s conditions

Joan Verdera

• Minimizers. . .

The candidate ellipse

If E is an ellipse then each even smooth homogeneous singular integral applied to χE is constant on E Then second order derivatives of the potential Pµ µ = χE |E| are constant on E Thus ∇Pµ is a linear function A(a, b)z + B(a, b)¯ z on E Take E centered at the origin and choose the semi-axis so that ∇Pµ = 0 on E: A(a, b) = 0, B(a, b) = 0 Hence Pµ is constant on E and (EL1) holds a = √1 − α b = √1 + α

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• Minimizers. . .

Proof of the second Euler condition - 1

If E is the candidate ellipse, then one has to prove that Pµ(z) ≥ Cα, z ∈ C \ E, µ = χE |E| It is enough to show that ∇Pµ does not vanish on C\E This requires a precise computation of ∇Pµ on C\E ∇Pµ(z) = −1 ¯ z ∗ µ + α 2 1 z ∗ µ − z ¯ z2 ∗ µ

• + z

Joan Verdera

• Minimizers. . .

Proof of the second Euler condition - 2

Let E be an ellipse with semiaxis a and b. Then 1 π¯ z ∗ χE

• (z) =

   z − λ¯ z, z ∈ E 2ab h(¯ z), z / ∈ E h(z) = 1 z + √ z2 + c2 c2 = b2 − a2 λ = a − b a + b

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• Minimizers. . .

z ¯ z2 = 1 ¯ z ∗ 1 π 1 ¯ z2 Conjugate Beurling = primitive in z and then −¯ ∂ 1 π 1 ¯ z2 = −¯ ∂ 1 π 1 ¯ z

• Joan Verdera
• Minimizers. . .

Proof of the second Euler condition - 3

• − z

¯ z2 ∗ µ

• (z) = 2λh(¯

z) − 2h′(¯ z)(z − λ¯ z − 2abh(¯ z)) h′(¯ z) = − h(¯ z) N(¯ z) N(z) =

• z2 + c2

∇Pµ(z)N(¯ z) = t x2 a2 + t + y2 b2 + t = 1 ∇Pµ(z), N(z) = t ∇Pµ(z), iN(z) = 0

Joan Verdera

• Minimizers. . .

Conclusion

The fact that evolution of ellipses under the vorticity form of Euler’s equation is a rotation with constant angular velocity (Kirchhoff) and the fact that minimisers of the non local energies with anisotropic interaction potential Wα(z) = − log |z| + α x2 |z|2 , −1 < α < 1 are normalised characteristic functions of suitably chosen ellipses are due, in last analysis, to the fact that the Cauchy transform of the characteristic function of an ellipse is linear on the ellipse 1 πz ⋆ χE

• (z) = ¯

z − λz, z ∈ E, λ = a − b a + b.

Joan Verdera

• Minimizers. . .

Conclusion (continued)

For the minimisation problem we did indeed use the precise form of the Cauchy integral of the charactaeristic function of the ellipse also off the ellipse, namely, 1 π¯ z ∗ χE

• (z) =

   z − λ¯ z, z ∈ E 2ab h(¯ z), z / ∈ E h(z) = 1 z + √ z2 + c2 c2 = b2 − a2 λ = a − b a + b. But the expression off the ellipse can be avoided by an alternative non computational argument.

Joan Verdera

• Minimizers. . .

Thank you for your attention and... Happy and productive retirement John and Don !!

Joan Verdera

• Minimizers. . .