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

Minimizers of non local energies, and ellipses Joan Verdera Universitat Aut` onoma 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  ∂ t v ( z, t ) + ( v · ∇ ) v ( z, t ) = −∇ p ( z, t )     ( E ) div v = 0     v ( z, 0) = v 0 ( z ) v · ∇ = v 1 ∂ 1 + v 2 ∂ 2 Joan Verdera Minimizers. . .

Acceleration and force Particle trajectory : dz ( t ) /dt = v ( z ( t ) , t ) d 2 z dt 2 = ∂ t v ( z ( t ) , t ) + ∂ 1 v ( z ( t ) , t ) v 1 ( z ( t ) , t ) + . . . = ∂ t v ( z, t ) + ( v · ∇ ) v ( z, t ) Pressure and force � � force on blob V = − p� n dS = −∇ p dx ∂V V Joan Verdera Minimizers. . .

Incompressibility : the velocity field is divergence free � � 0 = v · � n dS = div( v ) dx ∂V V Joan Verdera Minimizers. . .

Vorticity ω = curl( v ) = ∂ 1 v 2 − ∂ 2 v 1 2 ∂v = 2 ∂ ∂z v = div v + i curl v = iω 1 ∂ = ∂ is the fundamental solution of πz ∂z � ω ( ζ, t ) v ( z, t ) = i 1 z ∗ ω = i ζ dA ( ζ ) z − ¯ 2 π ¯ 2 π ¯ Joan Verdera Minimizers. . .

The vorticity equation   ∂ t ω + ( v · ∇ ) ω = 0     v = i 1 1 z ∗ ω = ∇ ⊥ N ∗ ω, N = 2 π log | z |  2 π ¯     ω ( z, 0) = ω 0 ( z ) Particle trajectory : dz ( t ) /dt = v ( z ( t ) , t ) dω ( z ( t ) , t ) = ∂ t ω ( z ( t ) , t ) + ∂ 1 ω ( z ( t ) , t ) v 1 ( z ( t ) , t ) + . . . dt Joan Verdera Minimizers. . .

Yudovich’s Theorem The vorticity equation is well posed in L ∞ : For each ω 0 ∈ L ∞ c ( C ) there is a unique global ”weak” solution to the vorticity equation with initial condition ω 0 . “Proof” : Solve dX ( z, t ) = v ( X ( z, t ) , t ) , X ( z, 0) = z dt and set ω ( z, t ) = ω 0 ( X − 1 ( z, t )) or ω ( X ( z, t ) , t ) = ω 0 ( z ) Joan Verdera Minimizers. . .

Vortex patches ω 0 = χ D , D a domain ω ( z, t ) = χ D ( t ) ( z ) D ( t ) D Joan Verdera Minimizers. . .

The two known explicit examples If D = D (0 , 1) is the unit disc, then D t = D (0 , 1) , 0 < t, χ D (0 , 1) ( z ) is a steady solution to the vorticity equation If D 0 = { ( x, y ) : x 2 /a 2 + y 2 /b 2 < 1 } is an ellipse then ab D t = e i Ω t D 0 , Kirchhoff: 0 < t, Ω = ( a + b ) 2 (vortex) Joan Verdera Minimizers. . .

Why ellipses rotate ? If E = { ( x, y ) : x 2 /a 2 + y 2 /b 2 ≤ 1 } is an ellipse then � 1 � πz ⋆ χ E ( 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 � 1 � � 2Ω z − z ⋆ χ D ( z ) ,� τ ( z ) � = 0 , z ∈ ∂D π ¯ If D is an ellipse with semiaxis a and b the equation is � 2Ω z − ( 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 D 0 has the origin as center of mass D t = e i Ω t D 0 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 R 3 � x − y � F ( x ) = −∇ P µ ( x ) = | x − y | 3 dµ ( y ) R 3 The potential energy of a proof charge at x due to µ is � 1 P µ ( x ) = | x − y | dµ ( y ) R 3 The energy of a charge distribution µ �� 1 E ( µ ) = | x − y | dµ ( x ) dµ ( y ) R 3 × R 3 Joan Verdera Minimizers. . .

Classical Potential Theory in the plane � z − w � F ( z ) = −∇ P µ ( z ) = | z − w | 2 dµ ( w ) C � P µ ( z ) = − log | z − w | dµ ( w ) C �� E ( µ ) = − log | z − w | dµ ( z ) dµ ( w ) C × C Books by Carleson and Wermer Joan Verdera Minimizers. . .

Potential Theory in the plane with an external field One considers a confinement force (book by Saff and Totik) � z − w F ( z ) = −∇ P µ ( z ) = − 2¯ � ∂P µ ( z ) = | z − w | 2 dµ ( w ) − z C � � − log | z − w | dµ ( w ) + | z | 2 | w | 2 P µ ( z ) = + dµ ( w ) 2 2 C C �� � | z | 2 dµ ( z ) I ( µ ) = − log | z − w | dµ ( z ) dµ ( w ) + C × C 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 1 unit disc D : π χ D ( z ) The minimiser is unique. Then you proceed by a direct calculation or by symmetry plus harmonicity Joan Verdera Minimizers. . .

IDEALISED DISLOCATIONS Straight & parallel; edge Single slip, and single sign (all positive) y INTERACTION STRESS generated by a dislocation at 0 and acting on a dislocation x at z F ( z ) = − κ ∇ V ( z ) V ( z ) = − log | z | + x 2 | z | 2 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 �� � ( − log | z − w | + ( x − u ) 2 � F ( z ) = −∇ P µ ( z ) = ∇ | z − w | 2 ) dµ ( w ) − z C � � � � − log | z − w | + ( x − u ) 2 dµ ( w )+ | z | 2 | w | 2 P µ ( z )= 2 + dµ ( w ) | z − w | 2 2 C C �� � � � − log | z − w | + ( x − u ) 2 | z | 2 dµ ( z ) I ( µ )= dµ ( z ) dµ ( w )+ | z − w | 2 C × C Joan Verdera Minimizers. . .

M. G. Mora, L. Rondi, L. Scardia (2016) It was a long standing conjecture that the unique minimiser of � � �� � − log | z − w | + ( x − u ) 2 | z | 2 dµ ( z ) I ( µ ) = − dµ ( z ) dµ ( w )+ | z − w | 2 C × C was the semicircle law on the vertical axis � µ := δ 0 ⊗ 1 2 − y 2 χ [ − √ √ 2] ( y ) dy 2 , π Joan Verdera Minimizers. . .

The problem Take the interaction potential given by convolution with the kernel W ( z ) = W α ( z ) = − log | z | + α x 2 | z | 2 , − 1 ≤ α ≤ 1 coupled with the confinement term | z | 2 � � �� � − log | z − w | + α ( x − u ) 2 | z | 2 dµ ( z ) I α ( µ )= − dµ ( z ) dµ ( w )+ | z − w | 2 C × C Joan Verdera Minimizers. . .

Carrillo, Mateu, Mora, Rondi, Scardia, V The unique minimiser of the energy I α is the normalized characteristic function of the ellipse centered at the origin with semi-axis √ √ 1 − α and 1 + α 1 0.5 0 -0.5 -1 -1.5 -1 -0.5 0 0.5 1 1.5 Joan Verdera Minimizers. . .

Euler conditions � � � − log | z − w | + α ( x − u ) 2 ( W α ∗ µ )( z ) = dµ ( w ) | z − w | 2 C � P µ ( z ) = ( W α ∗ µ )( z ) + | z | 2 + 1 | w | 2 dµ ( w ) 2 2 C (EL1) P µ ( z ) = C α , Cap a.e. on the support of µ (EL2) P µ ( z ) ≥ C α , off 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 W α on test functions ϕ with 0 integral � � (1 − α ) ξ 2 1 + (1 + α ) ξ 2 ( W α )( ξ ) ϕ ( ξ ) dA ( ξ ) = 1 � 2 ϕ ( ξ ) dA ( ξ ) | ξ | 4 2 π � � � ν ( ξ ) | 2 dξ, ( W α ∗ ν ) dν = ( W α )( ξ ) | � ν (1) = 0 Strict convexity implies uniqueness of the minimiser and sufficiency of Euler’s conditions Joan Verdera Minimizers. . .

The candidate ellipse If E is an ellipse then each even smooth homogeneous singular integral applied to χ E is constant on E µ = χ E Then second order derivatives of the potential P µ | E | are constant on E Thus ∇ P µ 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 ∇ P µ = 0 on E : A ( a, b ) = 0 , B ( a, b ) = 0 Hence P µ is constant on E and (EL1) holds a = √ 1 − α b = √ 1 + α Joan Verdera Minimizers. . .

Proof of the second Euler condition - 1 If E is the candidate ellipse, then one has to prove that µ = χ E z ∈ C \ E, P µ ( z ) ≥ C α , | E | It is enough to show that ∇ P µ does not vanish on C \ E This requires a precise computation of ∇ P µ on C \ E � 1 � ∇ P µ ( z ) = − 1 z ∗ µ + α z ∗ µ − z z 2 ∗ µ + z ¯ 2 ¯ Joan Verdera Minimizers. . .

Proof of the second Euler condition - 2 Let E be an ellipse with semiaxis a and b . Then  � 1 �  z − λ ¯ z, z ∈ E z ∗ χ E ( z ) = π ¯  2 ab h (¯ z ) , z / ∈ E 1 λ = a − b c 2 = b 2 − a 2 √ h ( z ) = z 2 + c 2 a + b z + Joan Verdera Minimizers. . .

z 2 = 1 z z ∗ 1 1 z 2 ¯ ¯ π ¯ Conjugate Beurling = primitive in z and then − ¯ ∂ � 1 � 1 1 1 z 2 = − ¯ ∂ π ¯ π z ¯ Joan Verdera Minimizers. . .