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 INTERFACE Model - . - M KRUMMEL CLASSICAL CAPILLARITY 2016 THEORY .
TWO FOR DROPLETS MODELS FORMATION GAUSS Free 1 BASED ENERGY ON CIRAOLO 2015 SHARP M INTERFACE Model - . - M KRUMMEL CLASSICAL CAPILLARITY 2016 THEORY . ( GPL ) 2 GATES PENROSE Free ENERGY LEBOWITZ . . DIFFUSED INTERFACE MODEL STATISTICAL MECHANICS RELATIVE Of NONLOCAL ALLEN CAHN Free ENERGY - - M BASED CARLEN 2015 ON . - M FIGALLI MOONEY 2016 - .
PART ONE htn GAUSS EIIR FREE DROPLET REGION ENERGY No - lE1=M FIXED E CONTAINER VOLUME of = - -1 € g1x)dx PC E) SURFACE POTENTIAL = + = TENSION ENERGY - E =H%E ) PERIMETER OF
⇒ PART ONE ntn GAUSS EIIR Free DROPLET REGION ENERGY No - lE1=m FIXED E CONTAINER VOLUME of = - + € g1n)dx PC E) SURFACE POTENTIAL = + = TENSION ENERGY - E =H%E ) PERIMETER OF ' ) pie )=O|m%t ! Is SMALL m > > Seg Olm ) =
⇒ ⇒ ⇒ PART ONE htn GAUSS EIIR FREE DROPLET REGION ENERGY No - lE1=M FIXED E CONTAINER Volume of = - + € g1n)dx PCE ) SURFACE POTENTIAL = + = TENSION ENERGY ~ E =H%E ) PERIMETER OF ' ) pie )=O|m%t ! Is SMALL m > > Seg Olm ) = GLOBAL MINIMIZERS ALMOST ISOPERI METRIC ALMOST CONSTANT CRITICAL POINTS MEAN CURV .
⇒ GLOBAL MINIMIZERS ALMOST ISOPERIMETRIC ( mtnt ' ) PCE ) 1+0 z as - Ciso lE/%nti ) 1 ¢ / ISOPERIMETRIC ENERGY COMPARISON INEQUALITY BALLS WITH
⇒ GLOBAL MINIMIZERS ALMOST ISOPERI METRIC ( mtnt ' ) PCE ) 1+0 z 21 - " hnti ) IEI - Ciso : - IMPROVED tom xmiggn "fE±mYh¥ PCE ) ISOPERIMCTRY 21 mprateuios cig ) Fusco ,
⇒ GLOBAL MINIMIZERS ALMOST ISOPERIMCTRIC mtnt ' ) PCE ) 1+01 z 21 - Ciso lE/%nti ) IMPROVED 21 tclnlxmiggn "fE±mYh¥ PCE ) ISOPERIMCTRY cig ) .( , Fuscomprateuios ADDITIONAL C ? DE ARGUMENTS IS QUANTITAVELY VARIATDNAL CLOSE 10 ) FIGALLIM A To SPHERE ( 15 ) IN A CONTAINER M. MIHAILA -
⇒ ALMOST CONSTANT CRITICAL POINTS MEAN CURV . jtmde '=HEt÷erk% ⇒ =ocm%td CLCONST DE Hot MEAN CURV OF MEANCURV . . . = ^Pk÷ DEFICIT Hoax +1 ) IEI
⇒ ⇒ CRITICAL ALMOST CONSTANT MEAN CURV POINTS . jtmde '=HEt÷rk% ⇒ =ocm%td CLCONST DE Hot MEAN CURV OF CURV Mean . . . = ^Pk÷ DEFICIT Hoax +1 ) IEI c= Hfe ) ( & ALEXANDROVTHM the OE -=c SPHERE
⇒ ⇒ ⇒ ALMOST CONSTANT MEAN CURV CRITICAL POINTS . 's %dE'=HYtg±e - c= Hfe ) ( & 1Hc% ⇒ =oCmht ALEXANDROVTHM the OE -=c SPHERE EXT ) ¢ 5) BALL CIRAOD DE INT RADIUS VCZZONI >0 g - { Hf ⇐ =n= B= UNIT Hops BALL hd ( OE ,dBdx ) ) ECCN ,g .pl#)dcmdE )
⇒ ⇒ ALMOST CONSTANT MEAN CURV CRITICAL POINTS . EXT ) ( 5) BALL CIRAOW DE RADIUS VCZZONI INT >0 g - { Hf ⇐ =n= B= UNIT Hops BALL hd ( OE ,0Blx ) ) ECCN ,g .pl#)dcmdE)EXT/1NT BALL RADIUS NOT CMC >0 ON TRUE ALMOST f Oh dcmdt ) BALLOF ~~ Radius , SMALL
⇒ ⇒ ALMOST CONSTANT MEAN CURV CRITICAL POINTS . )E( .( 15 ) Lte ) PIB ) LEN Hfe=h PIE CIRAOLOM 0< 2<1 ⇐ mdE)E8dh 't 't ) G ff uea union UNHRADNS of C " × ZE G QUANTITAVELY TO TANGENT CLOSE BALLS 1 2 74 lE,q¥eC(n)L£cmdE ) & MANY E.G OTHER . estimates .
⇒ ⇒ ALMOST CONSTANT MEAN CURV CRITICAL POINTS . 1+e)P( PIEIE ( " Hfe=h B) Krummel 0<24 'M { go.my#q8dni2 ' ) oE=(H+uH)x , } Huller :n£0B ECHHOFMDEI
⇒ ⇒ ALMOST CONSTANT MEAN CURV CRITICAL POINTS . 1+e PIEIE ( ) PIB ) " H8e=h Krummel 0<24 'M { go.my#q8dni2 ' ) oE=(H+uH)x:n£0B , } Huller ECHHOFMDEI M£3 ) ( AT IN FACT We CAN Use LEAST IF o£o#emmHH÷ettt% ⇒ ktt÷eFk% ⇒ }
⇒ ⇒ ALMOST CONSTANT MEAN CURV CRITICAL POINTS . (1+2) PIEIE PIB ) " Hfe=h Krummel 0<24 'M { go.my#q8dni2 ' ) oE=(H+uH)x:n£0B , } Huller ECHHOFMDEI M£3 ) ( AT IN FACT we CAN Use LEAST IF :# :,ktt÷eFk% ⇒ } o£o#emmlkth÷ . RELATED ALMGREN TO PRINCIPLE ISOPERIMETRIC
ALMGREN PRINCIPLE ISOPERIMETRIC ) ( 1 CODIMENSION VERSION PIEIZPCBD Ha IF THEN En
⇒ ALMGREN PRINCIPLE ISOPERIMETRIC PIE ) >_ PCBD IF Ho ,[ THEN En @ " ) , )=7iC$ - Soaldettual PCB .
⇐ ALMGREN PRINCIPLE ISOPERIMETRIC PCEIZPCBD IF Ho ,[ THEN En @ " ) , )=7iC$ PCB - Soaldettual ' fakoatfnoekoa ' .
ALMGREN PRINCIPLE ISOPERIMETRIC PIEIZPCBD IF Ho ,[ THEN En @ " ) , )=7iC$ PCB - Soaldettual =hakoa=!anoekoA (@a " take 'oaH
ALMGREN PRINCIPLE ISOPERIMETRIC IF Ho ,[ THEN En PlEIzPCBD@PCBdttiCsY-SoaldetFua1-Soakoa-fa.n !xoA (@a " take 'oaH QH "(oAnoE)eH%E)=P( E)
ALMGREN PRINCIPLE ISOPERIMETRIC IF Ho ,[ THEN En ekoa PlEIzPCBD@PCBdttiCsY-Soa1detTua1-Soakoa-fa.o (@a " take 'oaH H%AnoE)eH%E)=P( I E) RMKI EQUALITY HOLDS E=Bdn) E ⇒
ALMGREN PRINCIPLE ISOPERIMETRIC IF Ho ,[ THEN En PlEIzPCBD@PCBdttiCsY-Soa1dettua1-Soakoa-fanoekoAQ-aefanfettoaHnQHYoAnoEIeHYoEtPcEjRMk1EQUAL1TY.H RMKZ Yes IT OLDS REMINDS ! E=B^ln) ! ABP E ⇒ A Lot
µ@ d(E)=PCE ) PCBDI H#En & KRUMMEL 16 IF foln ) SMALL M - - 2E=dDu . THEN dr E )Ec(n)oCE " ) " D PCD Dust set 2h CONNECTED @
%y d(E)=PCE ) PCBDI H#En & KRUMMEL 16 IF foln ) SMALL M - - 2E=dDu . THEN dr E tent " ) " D PCD Dust set )Ec(n)dCE b 2r CONNECTED .HN/or*nr)accn)dcE @ r±r* Hon .*lEn §£ " )
@b d(E)=PCE ) PCBDI H#En & KRUMMEL 16 IF foln ) SMALL M - - . THEN E )d( 2EedDudr@o.D " JIE ) " PC D) DUST Ean ) set 2R CONNECTED IHor*lEn rent Hirlttlnldrtiahaccn #A * punwntuutllaacksdk E) 2r*={ ( } tuna :xe0Bz " u "a±" ( b " ' - 1h23 n=l Y¥eeg( coyote ) n=2 { ( Eph
PARTTNO GPL Free ENERGY ~ µJ(r ) NIT " → " 1.1 ) C- F §nU=mEf4D r - 0 flu )=fS . Wy )PJ( in . ybdndy Iucn ) 1- § nwluk TyTn Nn ) )dx Him Wtu÷
PART GPL TWO Free ENERGY ~ µJ( r ) T " → " 1.1 ) C- F ni §nU=mEf1 , 1) r - O flu )=fS - Wy )PJ( in . yDdndy Iucn ) TyTn - +§nW1uK))dr A Wtu ) SMALE MA & IF U ENERGY Low =|^#Ez THEN UCN ) I SHARP TRANSITION ALONG A SPHERE SMALL
PART GPL TWO Free ENERGY ^ µJ( r ) T " → " 1.1 ) C- F Ni §nU=mEf1 , 1) r - O )= ff - Wy )PJ( in . yDdndy flu Iucn ) + § nwluk TyTn ÷ ) )dx Wfu ) * MA & SMALL IF U ENERGY LOW =|^#[z THEN UCN ) T SHARP TRANSITION ALONG A SPHERE SMALL ENERGY OBSERVED LOW Be LIKELY To = * BUT NOT MUCH CARLEN TOO OTHERWISE UNIFORM See SURVEY STATE i ...
⇒ PART GPL TWO Free ENERGY % UIT " → §nU=mEf4D 1.1 ) C- STNJ =1 Flu )=$ . ybdndytfgnwlulx - Wy )PJ( in Iucn ) ) )dx a- ' id Wtu± , EE .
→ " ) F " → R ( PART GPL TWO Free ENERGY Ninh S U=mEf4D 1.1 ) S T C- =L Rn en i € D÷ . ybdndytfwlu )= ffl ucntwyskan flu 'xDdr , Rn " RKR 't ) jfym U* Flu 2 SHWARTZREARRANG U . - . - BY RICSZ REARRANGEMENT INQ .
→ PART GPL TWO Free ENERGY Ninh = f S U=mEf4D 1.1 ) J C- =L Rn en d ⇐ D*÷h . ybdndytfwlu )= ffl ucnhwyikan flu 'xDdx Rn " RNXR 't ) f yen U* Flu 2 SHWARTZREARRANG U . BY RICSZ REARRANGEMENT INQ . RADIALLY CARLEN DECREASING CARVALHO ESPOSITO ! 1 ! - . ( 09 ) MARRA LEBOWITZ LOW - ENERGY STATES
→ PART GPL TWO Free ENERGY mill S melt 1.1 ) S J C- ,D =L U= Rn # n € µ"÷dd . ybdndytfwlu )= ffl ucnhuiyikan flu 'xDdx Rn " RKR 't ) f syn U* Flu 2 SHWARTZREARRANG U . - r - By Rlesz REARRANGEMENT INQ . RADIALLY CARLEN DECREASING CARVALHO ESPOSITO ! 1 ! - . ( 09 ) MARRA LEBOWITZ LDW - ENERGY STATES . ( 15 ) CARLEN M - QUANTITATIVE Riesz ! 2 ! BASED ON A QUANTITATIVE REARRANGEMENT INQ BRUNN INEQUALITY MWKOWSII -
→ PART GPL TWO Free ENERGY mill f S U=mEf1,D 1.1 ) J C- =L Rn # n € µ"÷dd . ybdndytfwlu ffl ucnhuiyskan flu )= 'xDdx Rn " RKR 't ) f syn U* Flu 2 SHWARTZREARRANG U . - r - 'S BY RICSZ REARRANGEMENT Survey CARLEN We . RADIALLY CARLEN DECREASING CARVALHO FOR ESPOSITO MORE ! 1 ! - . GPL ( 09 ) MARRA LEBOWITZ LDW - ENERGY STATES . ( 15 ) CARLEN M - QUANTITATIVE Riesz ! 2 ! BASED ON A QUANTITATIVE REARRANGEMENT INQ BRUNN INEQUALITY MWKOWSII -
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