4/7/16 TRIESTE ICTP & CALCVAROPTTRAN GMT LYON U . TWO - - PowerPoint PPT Presentation

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4/7/16 TRIESTE ICTP & CALCVAROPTTRAN GMT LYON U . TWO - - PowerPoint PPT Presentation

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THEOREMS STABILITY & APPLICATIONS To PHASE TRANSITIONS F . MAGGI 4/7/16 TRIESTE ICTP & CALCVAROPTTRAN GMT LYON U . TWO FOR DROPLETS MODELS FORMATION GAUSS Free 1 BASED ENERGY ON CIRAOLO 2015 SHARP M

slide-1
SLIDE 1 STABILITY THEOREMS & APPLICATIONS To PHASE TRANSITIONS F . MAGGI ICTP TRIESTE CALCVAROPTTRAN & GMT U . LYON

4/7/16

slide-2
SLIDE 2 TWO MODELS FOR DROPLETS FORMATION 1 GAUSS Free ENERGY BASED ON SHARP INTERFACE Model CIRAOLO
  • M
. 2015 CLASSICAL CAPILLARITY THEORY KRUMMEL
  • M
. 2016
slide-3
SLIDE 3 TWO MODELS FOR DROPLETS FORMATION 1 GAUSS Free ENERGY BASED ON SHARP INTERFACE Model CIRAOLO
  • M
. 2015 CLASSICAL CAPILLARITY THEORY KRUMMEL
  • M
. 2016 2 GATES . PENROSE . LEBOWITZ ( GPL ) Free ENERGY DIFFUSED INTERFACE MODEL STATISTICAL MECHANICS NONLOCAL RELATIVE Of ALLEN
  • CAHN
Free ENERGY BASED ON CARLEN
  • M
. 2015 FIGALLI
  • M
.
  • MOONEY
2016
slide-4
SLIDE 4 PART ONE htn GAUSS FREE EIIR DROPLET REGION ENERGY
  • No
CONTAINER VOLUME
  • f
E = lE1=M FIXED
  • =
SURFACE + POTENTIAL = PC E)
  • 1 €g1x)dx
TENSION ENERGY
  • PERIMETER
OF E =H%E )
slide-5
SLIDE 5 PART ONE ntn GAUSS Free EIIR DROPLET REGION ENERGY
  • No
CONTAINER VOLUME
  • f
E = lE1=m FIXED
  • =
SURFACE + POTENTIAL = PC E) + €g1n)dx TENSION ENERGY
  • PERIMETER
OF E =H%E ) m Is SMALL ! ⇒ pie )=O|m%t ' ) > > Seg = Olm )
slide-6
SLIDE 6 PART ONE htn GAUSS FREE EIIR DROPLET REGION ENERGY
  • No
CONTAINER Volume
  • f
E = lE1=M FIXED
  • =
SURFACE + POTENTIAL = PCE ) + €g1n)dx TENSION ENERGY ~ PERIMETER OF E =H%E ) m Is SMALL ! ⇒ pie )=O|m%t ' ) > > Seg = Olm ) GLOBAL MINIMIZERS ⇒ ALMOST ISOPERI METRIC CRITICAL POINTS ⇒ ALMOST CONSTANT MEAN CURV .
slide-7
SLIDE 7 GLOBAL MINIMIZERS ⇒ ALMOST ISOPERIMETRIC 1+0 ( mtnt ' ) z PCE )
  • as
1 Ciso lE/%nti ) ENERGY

/

¢ ISOPERIMETRIC COMPARISON INEQUALITY WITH BALLS
slide-8
SLIDE 8 GLOBAL MINIMIZERS ⇒ ALMOST ISOPERI METRIC 1+0 ( mtnt ' ) z PCE )
  • 21
Ciso IEI " hnti )
  • :
  • IMPROVED
ISOPERIMCTRY PCE ) Fusco mprateuios ) cig , 21

tom

xmiggn "fE±mYh¥
slide-9
SLIDE 9 GLOBAL MINIMIZERS ⇒ ALMOST ISOPERIMCTRIC 1+01 mtnt ' ) z PCE )
  • 21
Ciso lE/%nti ) IMPROVED ISOPERIMCTRY PCE )

Fuscomprateuios

) cig , 21 tclnlxmiggn "fE±mYh¥ ADDITIONAL VARIATDNAL ARGUMENTS DE IS QUANTITAVELY C ? CLOSE FIGALLIM

.(

10) To A SPHERE IN A CONTAINER M.
  • MIHAILA
( 15 )
slide-10
SLIDE 10 CRITICAL POINTS ⇒ ALMOST CONSTANT MEAN CURV .

jtmde

'=HEt÷erk%⇒=ocm%td

CLCONST . MEANCURV . Hot MEAN CURV . OF DE DEFICIT Hoax = ^Pk÷ +1 ) IEI
slide-11
SLIDE 11 CRITICAL POINTS ⇒ ALMOST CONSTANT MEAN CURV .

jtmde

'=HEt÷rk%⇒=ocm%td

CLCONST . Mean CURV . Hot MEAN CURV . OF DE DEFICIT Hoax = ^Pk÷ +1 ) IEI ALEXANDROVTHM the
  • =c
⇒ OE SPHERE ( & c= Hfe )
slide-12
SLIDE 12 CRITICAL POINTS ⇒ ALMOST CONSTANT MEAN CURV . %dE'=HYtg±e
  • 1Hc%⇒=oCmht
's ALEXANDROVTHM the
  • =c
⇒ OE SPHERE ( & c= Hfe ) CIRAOD
  • VCZZONI
¢ 5) DE EXT ) INT BALL RADIUS g >0 { Hf⇐=n= Hops B= UNIT BALL ⇒ hd ( OE ,dBdx ) ) ECCN ,g .pl#)dcmdE )
slide-13
SLIDE 13 CRITICAL POINTS ⇒ ALMOST CONSTANT MEAN CURV . CIRAOW
  • VCZZONI
( 5) DE EXT ) INT BALL RADIUS g >0 { Hf⇐=n= Hops B= UNIT BALL ⇒ hd ( OE ,0Blx ) ) ECCN ,g

.pl#)dcmdE)EXT/1NT

BALL RADIUS f >0 NOT TRUE ON ALMOST CMC Radius ,

Oh

dcmdt ) BALLOF

~~

SMALL
slide-14
SLIDE 14 CRITICAL POINTS ⇒ ALMOST CONSTANT MEAN CURV . CIRAOLOM .( 15) Hfe=h PIE )E( Lte ) PIB ) LEN 0< 2<1

⇐mdE)E8dh

't 't) union
  • f
UNHRADNS ⇒ ZE QUANTITAVELY C " × CLOSE TO TANGENT BALLS G 1 2 74 & MANY OTHER E.G .

lE,q¥eC(n)L£cmdE

) estimates

Gffuea

.
slide-15
SLIDE 15 CRITICAL POINTS ⇒ ALMOST CONSTANT MEAN CURV . Hfe=h PIEIE (

1+e)P(

B) 0<24 Krummel 'M ' "

{ go.my#q8dni2

) ⇒
  • E=(H+uH)x
:n£0B , } Huller ECHHOFMDEI
slide-16
SLIDE 16 CRITICAL POINTS ⇒ ALMOST CONSTANT MEAN CURV . H8e=h PIEIE (

1+e

) PIB ) 0<24 Krummel 'M ' "

{ go.my#q8dni2

) ⇒
  • E=(H+uH)x:n£0B
, } Huller ECHHOFMDEI IN FACT We CAN Use ( AT LEAST IF M£3 )
  • £o#emmHH÷ettt%⇒ktt÷eFk%⇒}
slide-17
SLIDE 17 CRITICAL POINTS ⇒ ALMOST CONSTANT MEAN CURV . Hfe=h PIEIE

(1+2)

PIB ) 0<24 Krummel 'M ' "

{ go.my#q8dni2

) ⇒
  • E=(H+uH)x:n£0B
, } Huller ECHHOFMDEI IN FACT we CAN Use ( AT LEAST IF M£3 )
  • £o#emmlkth÷

:#

.

:,ktt÷eFk%⇒}

RELATED TO ALMGREN ISOPERIMETRIC PRINCIPLE
slide-18
SLIDE 18 ALMGREN ISOPERIMETRIC PRINCIPLE ( CODIMENSION 1 VERSION ) IF Ha En THEN PIEIZPCBD
slide-19
SLIDE 19 ALMGREN ISOPERIMETRIC PRINCIPLE IF Ho ,[ En THEN PIE ) >_ PCBD

@

PCB , )=7iC$ " )
  • Soaldettual

⇒

.
slide-20
SLIDE 20 ALMGREN ISOPERIMETRIC PRINCIPLE IF Ho ,[ En THEN PCEIZPCBD

@

PCB , )=7iC$ " )
  • Soaldettual

⇐

. ' ' fakoatfnoekoa
slide-21
SLIDE 21 ALMGREN ISOPERIMETRIC PRINCIPLE IF Ho ,[ En THEN PIEIZPCBD

@

PCB , )=7iC$ " )
  • Soaldettual
=hakoa=!anoekoA

(@a

take

'oaH

"
slide-22
SLIDE 22 ALMGREN ISOPERIMETRIC PRINCIPLE IF Ho ,[ En THEN PlEIzPCBD@PCBdttiCsY-SoaldetFua1-Soakoa-fa.n !xoA

(@a

take

'oaH

" QH "(oAnoE)eH%E)=P( E)
slide-23
SLIDE 23 ALMGREN ISOPERIMETRIC PRINCIPLE IF Ho ,[ En THEN PlEIzPCBD@PCBdttiCsY-Soa1detTua1-Soakoa-fa.o ekoa

(@a

take

'oaH

" I H%AnoE)eH%E)=P( E) RMKI EQUALITY HOLDS E⇒ E=Bdn)
slide-24
SLIDE 24 ALMGREN ISOPERIMETRIC PRINCIPLE IF Ho ,[ En THEN PlEIzPCBD@PCBdttiCsY-Soa1dettua1-Soakoa-fanoekoAQ-aefanfettoaHnQHYoAnoEIeHYoEtPcEjRMk1EQUAL1TY.H OLDS RMKZ Yes ! IT REMINDS E⇒ E=B^ln) A Lot ABP !
slide-25
SLIDE 25 KRUMMEL
  • M
. 16 IF H#En & d(E)=PCE )
  • PCBDI
foln ) SMALL E THEN

2E=dDu

dr

µ@

D " Dust set " PCD

)Ec(n)oCE

)

@

2h CONNECTED
slide-26
SLIDE 26 KRUMMEL
  • M
. 16 IF H#En & d(E)=PCE )
  • PCBDI
foln ) SMALL E THEN

2E=dDu

dr

%y

D " Dust set " PCD

)Ec(n)dCE

) b 2r CONNECTED

@

r±r* Hon .*lEn

§£

"

tent

.HN/or*nr)accn)dcE

)
slide-27
SLIDE 27 KRUMMEL
  • M
. 16 IF H#En & d(E)=PCE )
  • PCBDI
foln ) SMALL E THEN

2EedDudr@o.D

" DUST set " PC D) Ean ) JIE ) 2R CONNECTED rent IHor*lEn

@b

*

#A

Hirlttlnldrtiahaccn

)d(

E) 2r*={ ( tuna :xe0Bz } n=l

punwntuutllaacksdk

' " u "a±" "

( b

Y¥eeg( coyote ) n=2 { (Eph
  • 1h23
slide-28
SLIDE 28 PARTTNO GPL Free ENERGY ~ F " NIT "→ C- 1.1 )

µJ(r

) §nU=mEf4D
  • r
flu )=fS Iucn ) . Wy )PJ( in . ybdndy TyTn Nn 1- § nwluk ) )dx

Wtu÷

Him

slide-29
SLIDE 29 PART TWO GPL Free ENERGY ~ F " ni T "→ C- 1.1 )

µJ(

r ) §nU=mEf1 , 1)
  • r
O flu )=fS Iucn )
  • Wy )PJ( in
. yDdndy TyTn
  • A
+§nW1uK))dr Wtu ) IF MA SMALE & U Low ENERGY THEN UCN ) I SHARP TRANSITION ALONG

=|^#Ez

A SMALL SPHERE
slide-30
SLIDE 30 PART TWO GPL Free ENERGY ^ F " Ni T "→ C- 1.1 )

µJ(

r ) §nU=mEf1 , 1)
  • r
O flu )= ff Iucn )
  • Wy )PJ( in
. yDdndy TyTn ÷ + § nwluk ) )dx Wfu ) * IF MA SMALL & U LOW ENERGY THEN UCN ) T SHARP TRANSITION ALONG

=|^#[z

A SMALL SPHERE LOW ENERGY = LIKELY To Be OBSERVED * BUT NOT TOO MUCH ... OTHERWISE UNIFORM STATE i See CARLEN SURVEY
slide-31
SLIDE 31 PART TWO GPL Free ENERGY

%

UIT "→ C- 1.1 ) §nU=mEf4D STNJ =1 a- Flu )=$ Iucn )
  • Wy )PJ( in
. ybdndytfgnwlulx ) )dx

Wtu±

id

, ' ⇒

EE

.
slide-32
SLIDE 32 PART TWO GPL Free ENERGY ( F "→R " )

Ninh

→ C- 1.1 ) S U=mEf4D S T =L Rn en

i€D÷

, flu )= fflucntwyskan . ybdndytfwlu 'xDdr RKR " Rn

jfym

. 2 Flu 't ) U* SHWARTZREARRANG . U
  • BY
RICSZ REARRANGEMENT INQ .
slide-33
SLIDE 33 PART TWO GPL Free ENERGY

Ninh

→ C- 1.1 ) S U=mEf4D f J =L Rn en d⇐D*÷h flu )= fflucnhwyikan . ybdndytfwlu 'xDdx RNXR " Rn

fyen

2 Flu 't ) U* SHWARTZREARRANG . U

=

BY RICSZ REARRANGEMENT INQ . !1! RADIALLY DECREASING CARLEN
  • CARVALHO
. ESPOSITO LOW ENERGY STATES LEBOWITZ
  • MARRA
( 09 )
slide-34
SLIDE 34 PART TWO GPL Free ENERGY

mill

→ C- 1.1 ) S U= melt ,D S J =L Rn # n€µ"÷dd flu )= fflucnhuiyikan . ybdndytfwlu 'xDdx RKR " Rn

fsyn

2 Flu 't ) U* SHWARTZREARRANG . U
  • r
By Rlesz REARRANGEMENT INQ . !1! RADIALLY DECREASING CARLEN
  • CARVALHO
. ESPOSITO LDW ENERGY STATES LEBOWITZ
  • MARRA
( 09 ) !2! QUANTITATIVE Riesz CARLEN
  • M
. ( 15 ) BASED ON A QUANTITATIVE REARRANGEMENT INQ BRUNN
  • MWKOWSII
INEQUALITY
slide-35
SLIDE 35 PART TWO GPL Free ENERGY

mill

→ C- 1.1 ) S U=mEf1,D f J =L Rn # n€µ"÷dd flu )= fflucnhuiyskan . ybdndytfwlu 'xDdx RKR " Rn

fsyn

2 Flu 't ) U* SHWARTZREARRANG . U
  • r
BY RICSZ REARRANGEMENT We . CARLEN 'S Survey !1! RADIALLY DECREASING CARLEN
  • CARVALHO
. ESPOSITO FOR MORE GPL LDW ENERGY STATES LEBOWITZ
  • MARRA
( 09 ) !2! QUANTITATIVE Riesz CARLEN
  • M
. ( 15 ) BASED ON A QUANTITATIVE REARRANGEMENT INQ BRUNN
  • MWKOWSII
INEQUALITY
slide-36
SLIDE 36 BM INEQUALITY IETFIHZIEFFIFY " HEFIR " = E , F HOMOTHETIC CONVEX Sets
slide-37
SLIDE 37 BM INEQUALITY IETFFNZIEFFIFPH HEFIR " = E , F HOMOTHETIC CONVEX Sets auantnatwern
  • #a=ma×H¥
, .tn#lHeEIEfrIn
  • 1)
HE ,f)=inf{ IEO.len.tt#s.t.1XFI=lEY
slide-38
SLIDE 38 BM INEQUALITY lEtFFnz1EFF1FFVE.FcRn-E.FHOMoTHeTlCcoNVeXSeTsauantnatweBmoee.a-maxfEeT.hITIHIjtnEfrIn-HNE.fI_infflE0gegthIs.t.1XFI-lEY@F1GALLlM.PRATeLL1E.F CONVEX d( E , F) 2 un ) X( E ,F ) ?
slide-39
SLIDE 39 BM INEQUALITY lEtFThzlEFF1FFVE.FcRn-E.FHOMoTHeTlCcoNVeXSeTsauantnatweBmoee.a-mayEeT.hETIhlIjItnEFI-HNE.fI_infflE0gegthIs.t.1XFI-lEY@F1GALL1M.PRATeLL1E.F CONVEX d( E , F) >_ cm X( E ,F ) ? !2! FIGALLITERISON EIF GENERIC NON . SHARP ESTIMATES CHRIST
slide-40
SLIDE 40 BM INEQUALITY lEtFFnz1EFF1FFVE.FcRn-E.FHOMoTHeTlCcoNVeXSeTsauantnatweBmoee.a-mayEeT.FfTIhlIjItnEFI-HNE.fI-infflE0geythIs.t.1XFI-lEY@F1GALL1M.PRATeLL1E.F CONVEX d( E , F) >_c(n)X( E ,F ) ? FIGALLITERISON E ,F GENERIC NON . SHARP ESTIMATES !2! CHRIST !3! CARLEN
  • M
. E GENERIC 8ft ,f ) maxfl .fF¥ , ]4+÷2c(n)Xf[,F)4 F CONVEX
slide-41
SLIDE 41 E GENERIC F
  • B
, r >o ⇒ BM Becomes EUCLIDEAN CONCENTRATION IIRCE )|2lIr( By )| tr >0 INEQUALITY z(€)={ Dist FROM E < t }
slide-42
SLIDE 42 E GENERIC F = B , r >0 ⇒ BM Becomes EUCLIDEAN CONCENTRATION IIRCE )|2lIr( By )| tr >0 INEQUALITY z(€)={ Dist FROM E < t } SHARP QUANTITATIVE f , a- ALL , . m . . MOONEY 46 ) EUCLIDEAN CONCENTRATION INEQUALITY 14*12171%11 { rtecnminfrz ,1r}Y£o,B±T¥] }
slide-43
SLIDE 43 E GENERIC F
  • B
, r >0 ⇒ BM Becomes EUCLIDEAN CONCENTRATION LIVE )|2lIr( Bre )| tr >0 INEQUALITY q(€)={ Dist FROM E < t } SHARP QUANTITATIVE f , gay , . m . . MOONEY 46 ) EUCLIDEAN CONCENTRATION INEQUALITY

1Ir#lzlIrlB⇐H{

rtecnminfrz ,1r}Y£o,B±T¥] } * NO SYMMETRIZATDN ( FUSOMDCATELLD
slide-44
SLIDE 44 E GENERIC F=B , r >0 ⇒ BM Becomes EUCLIDEAN CONCENTRATION LIVE )|2lIr( By )| tr >0 INEQUALITY q(€)={ Dist FROM E < r } SHARP QUANTITATIVE f , gay , . m . . MOONEY 46 ) EUCLIDEAN CONCENTRATION INEQUALITY

1Ir#lzlIrlB⇐H{

rtecnsminfrz ,1r}Y£o,B±T¥] } * NO SYMMETRIZATDN * NO MASS TRANSP . ( FUSOMDCATELLD ( FIEALYMPRATELLD
slide-45
SLIDE 45 E GENERIC F=B , r >0 ⇒ BM Becomes EUCLIDEAN CONCENTRATION LIVE )|2lIr( By )| tr >0 INEQUALITY z(€)={ Dist FROM E < r } SHARP QUANTITATIVE fiaaytm . . MOONEY 46 ) EUCLIDEAN CONCENTRATION INEQUALITY 14*12171%11 { rtecnminfrz ,1r}Y£o,B±,t¥] } * NO SYMMETRIZATDN * NO MASS TRANSP . * NO REGUALIRITY ( FUSOMDCATELLD ( FIGALYMPRATELLD ( CICALESE LEONARD ) )