CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS
A series of lectures on topics of current research interest in applied mathematics under the direction of the Conference Board of the Mathematical Sciences, supported by the National Science Foundation and published by SIAM.
Garrett Birkhoff, The Numerical Solution of Elliptic Equations
D. V. Lindley, Bayesian Statistics, A Review
R. S. Varga, Functional Analysis and Approximation Theory in Numerical Analysis
R. R. Bahadur, Some Limit Theorems in Statistics
Patrick Billingsley, Weak Convergence of Measures: Applications in Probability
J. L. Lions, Some Aspects of the Optimal Control of Distributed Parameter Systems
Roger Penrose, Techniques of Differential Topology in Relativity
Herman Chernoff, Sequential Analysis and Optimal Design
J. Durbin, Distribution Theory for Tests Based on the Sample Distribution Function
Sol I. Rubinow, Mathematical Problems in the Biological Sciences
P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves
I. J. Schoenberg, Cardinal Spline Interpolation
Ivan Singer, The Theory of Best Approximation and Functional Analysis
Werner C. Rheinboldt, Methods of Solving Systems of Nonlinear Equations
Hans F. Weinberger, Variational Methods for Eigenvalue Approximation
R. Tyrrell Rockafellar, Conjugate Duality and Optimization
Sir James Lighthill, Mathematical Biofluiddynamics
Gerard Salton, Theory of Indexing
Cathleen S. Morawetz, Notes on Time Decay and Scattering for Some Hyperbolic Problems
F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epidemics
Richard Askey, Orthogonal Polynomials and Special Functions
L. E. Payne, Improperly Posed Problems in Partial Differential Equations
S. Rosen, Lectures on the Measurement and Evaluation of the Performance of Computing Systems
Herbert B. Keller, Numerical Solution of Two Point Boundary Value Problems
J. P. LaSalle, The Stability of Dynamical Systems
D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications
Peter J. Huber, Robust Statistical Procedures
Herbert Solomon, Geometric Probability
Fred S. Roberts, Graph Theory and Its Applications to Problems of Society
Juris Hartmanis, Feasible Computations and Provable Complexity Properties
Zohar Manna, Lectures on the Logic of Computer Programming
Ellis L. Johnson, Integer Programming: Facets, Subadditivity, and Duality for Group and Semi-group Problems
Shmuel Winograd, Arithmetic Complexity of Computations
J. F. C. Kingman, Mathematics of Genetic Diversity
Morton E. Gurtin, Topics in Finite Elasticity
Thomas G. Kurtz, Approximation of Population Processes
Jerrold E. Marsden, Lectures on Geometric Methods in Mathematical Physics
Bradley Efron, The Jackknife, the Bootstrap, and Other Resampling Plans
M. Woodroofe, Nonlinear Renewal Theory in Sequential Analysis
D. H. Sattinger, Branching in the Presence of Symmetry
R. Temam, Navier–Stokes Equations and Nonlinear Functional Analysis
Miklós Csörg, Quantile Processes with Statistical Applications
J. D. Buckmaster and G. S. S. Ludford, Lectures on Mathematical Combustion
R. E. Tarjan, Data Structures and Network Algorithms
Paul Waltman, Competition Models in Population Biology
S. R. S. Varadhan, Large Deviations and Applications
Kiyosi Itô, Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces
Alan C. Newell, Solitons in Mathematics and Physics
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Pranab Kumar Sen, Theory and Applications of Sequential Nonparametrics
László Lovász, An Algorithmic Theory of Numbers, Graphs and Convexity
E. W. Cheney, Multivariate Approximation Theory: Selected Topics
Joel Spencer, Ten Lectures on the Probabilistic Method
Paul C. Fife, Dynamics of Internal Layers and Diffusive Interfaces
Charles K. Chui, Multivariate Splines
Herbert S. Wilf, Combinatorial Algorithms: An Update
Henry C. Tuckwell, Stochastic Processes in the Neurosciences
Frank H. Clarke, Methods of Dynamic and Nonsmooth Optimization
Robert B. Gardner, The Method of Equivalence and Its Applications
Grace Wahba, Spline Models for Observational Data
Richard S. Varga, Scientific Computation on Mathematical Problems and Conjectures
Ingrid Daubechies, Ten Lectures on Wavelets
Stephen F. McCormick, Multilevel Projection Methods for Partial Differential Equations
Harald Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods
Joel Spencer, Ten Lectures on the Probabilistic Method, Second Edition
Charles A. Micchelli, Mathematical Aspects of Geometric Modeling
Roger Temam, Navier–Stokes Equations and Nonlinear Functional Analysis, Second Edition
Glenn Shafer, Probabilistic Expert Systems
Peter J. Huber, Robust Statistical Procedures, Second Edition
J. Michael Steele, Probability Theory and Combinatorial Optimization
Werner C. Rheinboldt, Methods for Solving Systems of Nonlinear Equations, Second Edition
J. M. Cushing, An Introduction to Structured Population Dynamics
Tai-Ping Liu, Hyperbolic and Viscous Conservation Laws
Michael Renardy, Mathematical Analysis of Viscoelastic Flows
Gérard Cornuéjols, Combinatorial Optimization: Packing and Covering
Irena Lasiecka, Mathematical Control Theory of Coupled PDEs
J. K. Shaw, Mathematical Principles of Optical Fiber Communications
Zhangxin Chen, Reservoir Simulation: Mathematical Techniques in Oil Recovery
Athanassios S. Fokas, A Unified Approach to Boundary Value Problems
Margaret Cheney and Brett Borden, Fundamentals of Radar Imaging
Fioralba Cakoni, David Colton, and Peter Monk, The Linear Sampling Method in Inverse Electromagnetic Scattering
Adrian Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis
Wei-Ming Ni, The Mathematics of Diffusion
Arnulf Jentzen and Peter E Kloeden, Taylor Approximations for Stochastic Partial Differential Equations
Fred Brauer and Carlos Castillo-Chavez, Mathematical Models for Communicable Diseases
Peter Kuchment, The Radon Transform and Medical Imaging
Roland Glowinski, Variational Methods for the Numerical Solution of Nonlinear Elliptic Problems
Bengt Fornberg and Natasha Flyer, A Primer on Radial Basis Functions with Applications to the Geosciences
Fioralba Cakoni, David Colton, and Houssem Haddar, Inverse Scattering Theory and Transmission Eigenvalues
Mike Steel, Phylogeny: Discrete and Random Processes in Evolution
Peter Constantin, Analysis of Hydrodynamic Models
Donald G. Saari, Mathematics Motivated by the Social and Behavioral Sciences
Yuji Kodama, Solitons in Two-Dimensional Shallow Water
Douglas N. Arnold, Finite Element Exterior Calculus
Qiang Du, Nonlocal Modeling, Analysis, and Computation
Alain Miranville, The Cahn–Hilliard Equation: Recent Advances and Applications
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Alain Miranville
Université de Poitiers
Poitiers, France
Chapter1 Introduction
TheCahn–Hilliardsystem
isusuallyrewritten,equivalently,asthefourth-order-in-spaceparabolicequation
whichispreciselytheequationknownastheCahn–Hilliardequation.Itwasproposed byJ.W.Cahn1 andJ.E.Hilliardin1958(see[81]).Theseequationsplayanessential roleinmaterialsscienceanddescribeimportantqualitativefeaturesoftwo-phasesystemsrelatedtophaseseparationprocesses,assumingisotropyandaconstanttemperature.Thiscanbeobserved,e.g.,whenabinaryalloy(e.g.,aluminum/zinc(see[467]) oriron/chromium(see[364,365,366]))iscooleddownsufficiently.Wethenobservea partialnucleation(i.e.,theappearanceofnuclidesinthematerial)oratotalnucleation, knownasspinodaldecomposition:theinitiallyhomogeneousmaterialquicklybecomes inhomogeneous,resultinginaveryfinelydispersedmicrostructure.Inasecondstage, whichoccursataslowertimescale,thesemicrostructurescoarsen(hencetheterm“coarsening”).See,e.g.,YouTube,https://www.youtube.com/watch?v=wWXS52OFo7w,foran animation.Suchphenomenaplayanessentialroleinthemechanicalpropertiesofthematerial,e.g.,strength,hardness,fracture,toughness,andductility.Wereferthereaderto, e.g.,[79,81,328,333,357,358,415,417]formoredetails.
Here, u istheorderparameter(wewillconsiderarescaleddensityofatomsorconcentrationofoneofthematerial’scomponentswhichtakesvaluesbetween 1 and 1,withthe values 1 and 1 correspondingtothepurestates.2 Thedensityofthesecondcomponentis u,meaningthatthetotaldensityisaconservedquantity3)and µ isthechemicalpotential (moreprecisely,thedifferenceinchemicalpotentialsbetweenthetwocomponents).Furthermore, f isthederivativeofadouble-wellpotential F .Athermodynamicallyrelevant
1JohnWernerCahn(January9,1928–March14,2016)playedamajorroleinmaterialsscience.
2Theorderparametervariescontinuouslythroughthe(diffuse)interfaceseparatingthepurestates,from 1 to 1
3If uA and uB denotethedensitiesofthetwocomponents,then,beforerescaling,wehave uA + uB =1 Replacing u by 2u 1,weobtain,afterrescaling, uA + uB =0.
∂u ∂t = κ∆µ,κ> 0,µ = α∆u + f (u),α> 0, (1.1)
∂u ∂t + ακ∆2 u κ∆f (u)=0, (1.2)
1 Downloaded 09/12/19 to 128.83.63.20. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Figure1.1. Comparisonbetweenregularandlogarithmicpotentials.CourtesyofShuiranPeng.
potential F isthefollowinglogarithmicfunction,whichfollowsfromamean-fieldmodel:
i.e.,
althoughthisfunctionisveryoftenapproximatedbyregularones,typically, F (s)= 1 4 (s2 1)2 (seeFigure1.1foracomparisonbetweenthetwopotentials),i.e., f (s)= s3 s; moregenerally,wecantake F (s)= 1 4 (s2 β2)2 , β ∈ R.Thelogarithmictermsin(1.3) correspondtotheentropyofmixing,and θ and θc areproportionaltotheabsolutetemperature(assumedconstantduringtheprocess)andacriticaltemperature,respectively;the condition θ<θc ensuresthat F hasadouble-wellformandthatphaseseparationcan occur.Alsonotethatthepolynomialapproximationisreasonablewhenthequenchisshallow,i.e.,whentheabsolutetemperatureisclosetothecriticaltemperature.Finally, κ is themobilityand α isrelatedtothesurfacetensionattheinterface.
Fromaphenomenologicalpointofview,theCahn–Hilliardsystemcanbederivedas follows.
Weconsiderthefollowing(total)freeenergy,calledtheGinzburg–Landaufreeenergy:
where Ω ⊂ Rn , n =1, 2,or 3,isthedomainoccupiedbythematerial.Thegradienttermin (1.5)isproposedin[81]tomodelthesurfaceenergyoftheinterface(i.e.,capillarity;note thatsuchgradientsgobacktoJ.D.vanderWaals[472]); F isalsocalledthehomogeneous freeenergy.
Wethenhavethemassbalance
where h isthemassflux,whichisrelatedtothechemicalpotential µ bythefollowing (postulated)constitutiveequationresemblingFick’slaw:
2Chapter1.Introduction s -1.5-1-0.500.511.5 F(s) -0.2 -0.1 0 0.1 0.2 0.3 0.4 PolynomialPotential LogarithmicPotential
F (s
θc 2 (1 s 2)+ θ 2 (1 s)ln 1 s 2 +(1+ s)ln 1+ s 2 ,s ∈ ( 1, 1), (1.3) 0 <θ<θc,
f
s
θcs + θ 2 ln 1+ s 1 s ,
)=
(
)=
(1.4)
ΨΩ(u, ∇u)= Ω α 2 |∇u|2 + F (u) dx, (1.5)
∂u ∂t = divh, (1.6)
h = κ∇µ. (1.7) Downloaded 09/12/19 to 128.83.63.20. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Theusualdefinitionofthechemicalpotentialisthatitisthederivativeofthefreeenergy withrespecttotheorderparameter.Here,thisdefinitionisincompatiblewiththepresence of ∇u inthefreeenergy.Instead, µ isdefinedasavariationalderivativeofthefreeenergy withrespectto u,whichyields(assumingproperboundaryconditions)
theCahn–Hilliardsystemthenfollows.Thisvariationalderivativecanbe(formally)seen bywritingthat,forasmallvariation,
where · denotestheusualEuclideanscalarproduct.Assumingcompatibleboundaryconditionsandintegratingbyparts,thisyields
fromwhichthedefinitionfollows.
TheCahn–Hilliardsystem,inaboundedandregulardomain Ω,isusuallyassociated withNeumannboundaryconditions,namely
meaningthatthereisnomassfluxattheboundary(notethat
whichisanaturalvariationalboundarycondition(by“natural,”wemeanthatwecan writeaconvenientvariational/weakformulationinviewofthemathematicalanalysisof theproblem;thisboundaryconditionalsoyieldsthattheinterfaceisorthogonaltothe boundary).Here, Γ= ∂Ω and ν istheunitouternormaltotheboundary.Inparticular,it followsfromthefirstboundaryconditionthatwehavetheconservationofmass,i.e.,ofthe spatialaverageoftheorderparameter,obtainedby(formally)integratingthefirstequation of(1.1)over Ω,
Ifwehaveinmindthefourth-order-in-spaceCahn–Hilliardequation,wecanrewritethese boundaryconditions,equivalently,as
Wecanalsoconsiderperiodicboundaryconditions(inwhichcase Ω=Πn i=1(0,Li), Li > 0, i =1,...,n);inthiscase,westillhavetheconservationofmass.Notethat wegenerallydonotconsiderDirichletboundaryconditions,preciselybecausetheydonot yieldtheconservationofmass,althoughsuchboundaryconditionscertainlysimplifythe mathematicalanalysis.
Now,thequestionofhowthephaseseparationprocess(i.e.,thespinodaldecomposition)isinfluencedbythepresenceofwallshasgainedmuchattention(see[202,203,317]
Chapter1.Introduction3
µ = α∆u + f (u); (1.8)
δΨΩ = Ω (α∇u ·∇δu + f (u)δu) dx,
δΨΩ = Ω ( α∆u + f (u))δudx,
∂µ ∂ν =0onΓ, (1.9)
h ν = κ ∂µ ∂ν ),and ∂u ∂ν =0onΓ, (1.10)
u(t) ≡ 1 Vol(Ω) Ω u(x,t) dx = u(0) ∀t ≥ 0. (1.11)
∂u ∂ν = ∂∆u ∂ν =0onΓ. (1.12)
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andthereferencestherein).Thisproblemisstudiedmainlyforpolymermixtures(although itshouldalsobeimportantforothersystems,suchasbinarymetallicalloys):fromatechnologicalpointofview,binarypolymermixturesareparticularlyinteresting,sincethe structuresoccurringduringthephaseseparationprocessmaybefrozenbyarapidquench intotheglassystate;microstructuresatsurfacesonverysmalllengthscalescanbeproducedinthisway.
Wealsorecallthattheusualvariationalboundarycondition ∂u ∂ν =0 yieldsthatthe interfaceisorthogonaltotheboundary,meaningthatthecontactline,whentheinterface betweenthetwocomponentsmeetsthewalls,isstatic,whichisnotreasonableinmany situations.Thisisthecase,e.g.,formixturesoftwoimmisciblefluids:inthiscase,the contactangleshouldbedynamic,duetothemovementsofthefluids.Thiscanalsobe thecaseinthecontextofbinaryalloys,meaningthatweneedtodefinedynamicboundary conditionsfortheCahn–Hilliardequation.
Inthiscase,weagainwritethatthereisnomassfluxattheboundary(i.e.,that(1.9)still holds).Then,toobtainthesecondboundarycondition,followingthephenomenological derivationoftheCahn–Hilliardsystem,weconsider,inadditiontotheusualGinzburg–Landaufreeenergyandassumingshort-rangeinteractionswiththewalls,asurfacefree energyoftheform
where ∇Γ isthesurfacegradientand G isasurfacepotential.Thus,thetotalfreeenergy ofthesystemreads
Writingfinallythatthesystemtendstominimizetheexcesssurfaceenergy,weareledto postulatetheboundarycondition
i.e.,thereisarelaxationdynamicsontheboundary.Thisboundaryconditionisusually referredtoasadynamicboundaryconditioninthesensethatthekinetics,i.e., ∂u ∂t ,appears explicitly.Here, ∆Γ istheLaplace–Beltramioperator, g = G ,and d> 0 issomerelaxationparameter.Furthermore,intheoriginalderivation,wehave G(s
s, where aΓ > 0 accountsforamodificationoftheeffectiveinteractionbetweenthecomponentsatthewallsand bΓ characterizesthepossiblepreferentialattraction(orrepulsion)of oneofthecomponentsbythewalls(when bΓ vanishes,thereisnopreferentialattraction). Wenotethatitfollowsfromtheboundaryconditionsthat,formally,
where · X denotesthenormontheBanachspace X,i.e.,thetotalfreeenergydecreases; inthecaseoftheclassicalNeumannboundaryconditions,wehave
Wealsoreferthereaderto[37,204]forotherphysicalderivationsofthedynamicboundary condition,obtainedbytakingthecontinuumlimitoflatticemodelswithinadirectmeanfieldapproximationandbyapplyingadensityfunctionaltheory,respectively;to[432]for
4Chapter1.Introduction
ΨΓ(u, ∇Γu)= Γ αΓ 2 |∇Γu|2 + G(u) dΣ,αΓ > 0, (1.13)
Ψ=ΨΩ +ΨΓ. (1.14)
1 d ∂u ∂t αΓ∆Γu + g(u)+ α ∂u ∂ν =0onΓ; (1.15)
)= 1 2 aΓs2 bΓ
dΨ dt = 1 κ ∂u ∂t 2 H 1(Ω) 1 d ∂u ∂t 2 L2(Γ) ≤ 0,
dΨΩ dt = 1 κ ∂u ∂t 2 H 1(Ω) ≤ 0.
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thederivationofdynamicboundaryconditionsinthecontextoftwo-phasefluidflows;and to[442,447]foranapproachbasedonconcentratedcapacity.
Actually,itwouldseemmorereasonable,inthecaseofnonpermeablewalls,towrite theconservationofmassbothinthebulk Ω andontheboundary Γ,i.e.,
Indeed,duetotheinteractionswiththewalls,weshouldexpectsomemassontheboundary.Weassumethatthefirstequationof(1.1)stillholds.Then,writingthat
where ∂ isthevariationalderivativementionedabove(notethat,intheoriginalderivation, wehave µ = ∂uΨΩ),weobtainthesecondequationof(1.1),togetherwiththeboundary condition
Wenownotethat,owingtothefirstequationof(1.1),theabovemassconservationreads
Aclassofboundaryconditionstoensurethismassconservationreads
Wecanthusseethat,when βΓ > 0,wealsohaveaCahn–Hilliard-typesystemonthe boundary.Notethatitfollowsfromtheabovethat
actually,inthecaseoftheusualNeumannboundaryconditions,wealsohave
Similardynamicboundaryconditions,inthecaseofsemipermeablewalls,areconsidered in[215,216,227].Furthermore,in[349],basedonanenergeticvariationalapproachand Onsager’sprincipleofmaximumenergydissipation,thesedynamicboundaryconditions arerecovered,togetherwiththeno-mass-fluxcondition(1.9);inthiscase,wehavemass conservationinthebulkandontheboundary,separately.
TheCahn–Hilliardsystem/equationisnowquitewellunderstood,atleastfromamathematicalpointofview.Inparticular,wehaveafairlycompletepictureasfarastheexistence,theuniqueness,andtheregularityofsolutionsandtheasymptoticbehaviorofthe associateddynamicalsystemareconcerned.Wereferthereaderto(amongahugeliterature),e.g.,[5,42,72,110,116,121,124,138,151,160,177,179,183,185,188,215, 216,227,242,250,251,257,316,331,340,349,352,355,383,399,400,404,410,411, 413,414,415,417,430,433,440,465,489,505].Asfarastheasymptoticbehaviorofthe systemisconcerned,wehave,inparticular,theexistenceoffinite-dimensionalattractors. Suchsetsgiveinformationontheglobal/allpossibledynamicsofthesystem.Furthermore, thefinitedimensionalitymeans,veryroughlyspeaking,that,eventhoughtheinitialphase
Chapter1.Introduction5
d dt Ω udx + Γ udΣ =0
µ = ∂uΨ,
µ = αΓ∆Γu + g(u)+ α ∂u ∂ν onΓ
Γ ∂u ∂t + κ ∂µ ∂ν dΣ=0
∂u ∂t = βΓ∆Γµ κ ∂µ ∂ν onΓ,βΓ ≥ 0.
dΨ dt = κ ∇µ 2 L2(Ω)n βΓ ∇Γµ 2 L2(Γ)n ≤ 0;
dΨΩ dt = κ ∇µ 2 L2(Ω)n ≤ 0
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spaceisinfinitedimensional,thelimitdynamicscanbedescribedbyafinitenumberof parameters.Werefertheinterestedreaderto,e.g.,[22,125,169,402,465]formoredetailsanddiscussionsonthis.Wealsohavetheconvergenceofsingletrajectoriestosteady states.
Wehavesofarassumedthatthemobility κ isapositiveconstant.Actually, κ isoften expectedtodependontheorderparameterandtodegenerateatthesingularpointsof f in thecaseofalogarithmicnonlinearterm(see[80,183,184,234,497];seealso[502]fora discussioninthecontextofimmisciblebinaryfluids).Note,however,thatthisessentially restrictsthediffusionprocesstotheinterfacialregionandisobserved,typically,inphysical situationsinwhichthemovementsofatomsareconfinedtothisregion(see[434]).Inthis case,thefirstequationof(1.1)reads
where,typically, κ(s)=1 s2.Inparticular,theexistenceofsolutionstotheCahn–Hilliardequationwithdegeneratemobilitiesandlogarithmicnonlinearitiesisprovedin [183];notethat,uptonow,onlytheexistenceofweaksolutionsisknown,andnothing else.Theasymptoticbehaviorand,moreprecisely,theexistenceofattractorsoftheCahn–Hilliardequationwithnonconstantandnondegeneratingmobilitiesarestudiedin[448, 451].
WecanalsonotethatthegradienttermintheGinzburg–Landaufreeenergy(1.5)accountsforthefactthatshort-rangeinteractionsbetweenthematerial’scomponentsare assumed.Actually,thistermisobtainedbyapproximatinganonlocaltermwhichalso accountsforlong-rangeinteractions(see[81]).Followingstochasticarguments,G.GiacominandJ.L.Lebowitzin[248,249]derivedtheCahn–Hilliardequation,withanonlocal term,byconsideringalatticegaswithlong-rangeKacpotentials(i.e.,theinteractionenergybetweentwoparticlesat x and y (x, y ∈ Zn)isgivenby γnK(γ|x y|), γ> 0 being senttozeroand K beingasmoothfunction).Inthiscase,the(total)freeenergyreads
where Tn isthe n-dimensionaltorus.Furthermore,rewritingthetotalfreeenergyinthe form
where k1(x)= Tn K(|x y|) dy,wecan,byexpandingthelasttermandkeepingonly sometermsintheexpansion,recovertheGinzburg–Landaufreeenergy(thisisreasonable whenthescaleonwhichthefreeenergyvariesislargecomparedwith γ 1;themacroscopicevolutionisobservedhereonthespatialscale γ 1 andtimescale γ 2);seealso [362].Suchmodelsarestudied,e.g.,in[3,31,214,217,218,219,223,304](seealso [104,162,277,278,279,341,494]forthenumericalanalysisandsimulations).
Now,itisinterestingtonotethattheCahn–Hilliardequationandsomeofitsvariantsarealsorelevantinphenomenaotherthanphaseseparationinbinaryalloys.We canmention,forinstance,dealloying(thiscanbeobservedincorrosionprocesses;see [190]);populationdynamics(see[129]);tumorgrowth(see[21,318]);bacterialfilms (see[326]);thinfilms(see[420,468]);chemistry(see[473]);imageprocessing(see [35,36,90,105,161]);astronomy,withtheringsofSaturn(see[471]);andecology(for
6Chapter1.Introduction
∂u ∂t =div(κ(u)∇µ),
ΨΩ(u)= Tn f (u(x))+ u(x) Tn K(|x y|)(1 u(y)) dy dx, (1.16)
ΨΩ(u)= Tn f (u(x))+ k1(x)u(x)(1 u(x))+ 1 2 Tn K(|x y|)|u(x) u(y)|2 dy dx,
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Chapter1.Introduction7
instance,theclusteringofmusselscanbeperfectlywelldescribedbytheCahn–Hilliard equation(see[351];seealsoYouTube,https://www.youtube.com/watch?v=u-mEjfBaYks andhttps://www.youtube.com/watch?v=OYcXZ7Ho4o8,forrealandsimulatedmussel clustering,respectively);ofcourse,inthiscase,thetimescaleismuchlarger,typically weeksormonths).
Inparticular,severalsuchphenomenacanbemodeledbythegeneralizedCahn–Hilliard equation
(here, α and κ donotnecessarilyhavethesamephysicalmeaningasintheoriginalCahn–Hilliardequation).Theabovegeneralequationcontains,inparticular,thefollowingmodels.
(i)MixedAllen–Cahn/Cahn–Hilliardsystem. Inthiscase,weconsiderthesystemof equations
whichcanberewritten,equivalently,as
andisindeedoftheform(1.17).Inparticular,withouttheterm
D∆µ inthefirstequation,wehavetheAllen–Cahnequation(whichdescribestheorderingofatomsduringthe phaseseparationprocess;see[13]),and,withouttheterm µ,wehavetheCahn–Hilliard equation.Theseequationsareproposedtoaccountformicroscopicmechanismssuchas surfacediffusionandadsorption/desorption,i.e.,adhesionofatomstoasurface/release ofasubstancefromorthroughasurface(see[310,312,313,363])andarestudiedin [300,301,302,303,311].
(ii)Cahn–Hilliard–Oono4 equation(see[372,418,474]). Inthiscase,
x,s)= g(s)= βs,β> 0.
Thisfunctionisproposedin[418]toaccountforlong-range(i.e.,nonlocal)interactions inphaseseparationandalsotosimplifynumericalsimulations,becausewedonothaveto accountfortheconservationofmass,althoughitseemsthatthisequationisnotconsidered insimulations.
Actually,itcanbesurprisingthatnonlocalinteractionscanbedescribedbysucha simplelinearterm.Thiscanbeseenbynotingthatweconsiderherethefreeenergy
wherethefunction g describesthelong-rangeinteractions.Inparticular,inOono’smodel andinthreespacedimensions,wetake
Thelong-rangeinteractionsarerepulsivewhen u(y) and u(x) havethesamesignandthus favortheformationofinterfaces(see[474]andthereferencestherein).Finally,asinthe 4AbetternamewouldbetheCahn–Hilliard–Oono–Puriequation;wewill,however,keepthecustomaryone.
∂u ∂t + ακ∆2 u κ∆f (u)+ g(x,u)=0,α,κ> 0 (1.17)
∂u ∂t = ε 2D∆µ µ,D,ε> 0,µ = ∆u + f (u) ε2 ,
∂u ∂t + ε 2D∆2 u ∆(Df (u)+ u)+ f (u) ε2 =0
ε
2
g
(
ΨΩ = Ω α 2 |∇u|2 + F (u)+ Ω u(y)k(x,y)u(x) dy dx, (1.18)
k(x,y)= β 4π|x y| (1.19)
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derivationoftheclassicalCahn–Hilliardequation,wehave
whichyieldstheCahn–Hilliard–Oonoequation,notingthat
y| istheGreenfunction associatedwiththeLaplaceoperator.Indeed,consideringagainasmallvariation,wehave
|
sothat
NotingthattheLaplaciancorrespondstothe x-variable,weseethat
Finally,bydefinitionofGreen’sfunctionanddenotingby di theDiracmassatzero(thisis ofcourseformal,sincetheDiracmassisnotafunction),wehave
whichyields
fromwhichtheCahn–Hilliard–Oonoequationfollows(with β replacedby κβ).This modelwillbeaddressedinChapter6(seealso[372,383]).
Avariantofthismodel,proposedin[123]tomodelmicrophaseseparationofdiblock copolymers,consistsoftaking
where u0 istheinitialcondition.Inthiscase,wehavetheconservationofmass;efficient simulationsareperformedin[20,102].ThisvariantoftheCahn–Hilliard–Oonoequation canalsobecoupledwiththeincompressibleNavier–Stokesequationstomodelachemicallyreactingbinaryfluid(see[296,297];seealso[59]forthemathematicalanalysis). (iii)Proliferationterm. Inthiscase,
Thisfunctionisproposedin[318]inviewofbiologicalapplicationsand,moreprecisely, tomodelwoundhealingandtumorgrowth(inonespacedimension;inthiscase,wecan thinkofapropagationfront)andtheclusteringofmalignantbraintumorcells(intwospace dimensions);seealso[473]forotherquadraticfunctionswithchemicalapplicationsand [21]forotherpolynomialswithbiologicalapplications.Thismodelwillbeaddressedin Chapter8(seealso[117,385]).
8Chapter1.Introduction
∂u ∂t = κ∆∂uΨΩ, (1.20)
1 4π
δΨΩ = Ω α∇u ·∇δu + f (u)δu + Ω k(x,y)u(y)δu(x) dy dx = Ω α∆u + f (u)+ Ω k(x,y)u(y) dy δu(x) dx
∂uΨΩ = α∆u + f (u)+ Ω k(x,y)u(y) dy.
x
∆∂uΨΩ = α∆2 u +∆f (u)+ Ω ∆k(x,y)u(y) dy.
Ω ∆(
β Ω di(x y)u(y) dy = βu(x),
∂uΨΩ = α∆u + f (u) βu,
k(x,y))u(y) dy =
g(x,s
g(s)= β s 1 Vol(Ω) Ω u0(x) dx ,β> 0,
)=
g
g(s)=
s 1),λ> 0
(x,s)=
λs(
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to
(iv)Fidelityterm. Inthiscase,
, where χ denotestheindicatorfunction,andweconsidertheequation
Writteninthisway, ε correspondstotheinterfacethickness.Thisfunction g isproposedin [35,36]inviewofapplicationstobinaryimageinpainting(i.e.,blackandwhiteimages). Here, h isagiven(damaged)imageand D istheinpainting(i.e.,damaged)region.Furthermore,thefidelityterm g(x,u) isaddedtokeepthesolution u closetotheimageoutside theinpaintingregion.Theideainthismodelistosolvetheequationuptosteadystateto obtainaninpainted(i.e.,restored)version u(x) of h(x).Thismodelwillbeaddressedin Chapter7(seealso[36,105,106]).
Thegeneralizedequation(1.17)isstudiedin[373,381](seealso[195])undervery generalassumptionsontheadditionalterm g,whenassociatedwithDirichletboundary conditions.Inthiscase,weessentiallyrecovertheresults(well-posedness,regularity,and existenceoffinite-dimensionalattractors)knownfortheoriginalCahn–Hilliardequation. ThecaseofNeumannboundaryconditionsismuchmoreinvolvedbecausewenolonger havetheconservationofmass,i.e.,ofthespatialaverageoftheorderparameter,when comparedwiththeoriginalCahn–HilliardequationwithNeumannboundaryconditions (see[105,106,117,195,196,241,385]).
AnothervariantoftheCahn–Hilliardequationisconcernedwithhigher-orderCahn–Hilliardmodels.Moreprecisely,G.CaginalpandE.Esenturkrecentlyproposedin[78] (seealso[98])higher-orderphase-fieldmodelstoaccountforanisotropicinterfaces(see also[327,464,484]forotherapproaches,which,however,donotprovideanexplicitway tocomputetheanisotropy).Moreprecisely,theseauthorsproposethefollowingmodified freeenergy,inwhichweomitthetemperature:
where(weconsiderherethecase n =3),for k =(k1,k2,k3) ∈ (N ∪{0
and,for k =(0, 0, 0),
(weagreethat D(0,0,0)v = v).Thecorrespondinghigher-orderCahn–Hilliardequation thenreads
For M =1 (anisotropicCahn–Hilliardequation),wehaveanequationoftheform
Chapter1.Introduction9
g
x,s
λ0χΩ\D(x)(s h(x)),λ0 > 0,D ⊂ Ω,h ∈ L2(Ω)
∂u ∂t + ε∆2 u 1 ε ∆f (u)+ g(x,u)=0,ε> 0.
(
)=
ΨHOGL = Ω 1 2 M i=1 |k|=i ak|Dk u|2 + F (u) dx,M ∈ N,ak > 0, |k| = M, (1.21)
})3 , |k| = k1 + k2 + k3
Dk = ∂|k| ∂xk1 1 ∂xk2 2 ∂xk3 3
∂u ∂t ∆ M i=1 ( 1)i |k|=i akD2k u ∆f (u)=0. (1.22)
∂u ∂t +∆ 3 i=1 ai ∂2u ∂x2 i ∆f (u)=0, Downloaded 09/12/19 to 128.83.63.20. Redistribution subject
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and,for M =2 (sixth-orderanisotropicCahn–Hilliardequation),wehaveanequationof theform
Westudyin[113,367]thecorrespondinghigher-orderisotropicmodel,namely
and,in[114],theanisotropichigher-ordermodel(1.22)(there,numericalsimulationsare alsoperformedtoillustratetheeffectsofthehigher-ordertermsandoftheanisotropy).Furthermore,thesemodelscontainsixth-orderCahn–Hilliardmodels.Wenotethatthereisa stronginterestinthestudyofsixth-orderCahn–Hilliardequations.Suchequationsarisein situationssuchas,e.g.,stronganisotropyeffectsinphaseseparationprocesses(see[470]), atomisticmodelsofcrystalgrowth(see[189,228]),thedescriptionofgrowingcrystalline surfaceswithsmallslopeswhichundergofaceting(see[447]),oil-water-surfactantmixtures(see[260,261]),andmixturesofpolymermolecules(see[150]).Wereferthereader to[95,269,273,274,276,293,329,330,359,360,372,377,379,380,382,425,426, 449,450,475,476,488]forthemathematicalandnumericalanalysisofsuchmodels.
Wecanalsonotethatthevariant(1.17)canberelevantinthecontextofhigher-order models(wecanmention,forinstance,anisotropiceffectsintumorgrowth).Wereferthe readerto[115]fortheanalysisandnumericalsimulationsofsuchmodels.
WefinallymentionseveralotherimportantgeneralizationsandvariantsoftheCahn–Hilliardequation.
ThefirstoneconsistsofstudyingsystemsofCahn–Hilliardequationstodescribephase separationinmulticomponentalloys(see[68,124,141,184,185,193,234,235,236, 395]).NotethattheCahn–Hilliardequationcanberewritten,equivalently,asasystemof two(Cahn–Hilliard)equations.Letusindeeddenoteby A and B thetwocomponentsand consider,withobviousnotation,thefreeenergy
Then,theCahn–Hilliardsystem(1.1)isequivalent,againwithobviousnotationandnoting that f isanoddfunctioninbothcasesofinterest,to
Furthermore,wecanseethat
10Chapter1.Introduction
∂u ∂t ∆ 3 i,j=1 aij ∂4u ∂x2 i ∂x2 j +∆ 3 i=1 bi ∂2u ∂x2 i ∆f (u)=0
∂u ∂t ∆P ( ∆)u ∆f (u)=0, (1.23) where P (s)= M i=1 ais i ,aM > 0,M ≥ 1,s ∈ R,
ΨΩ(uA, ∇uA,uB , ∇uB )= 1 2 Ω α 2 |∇uA|2 + α 2 |∇uB |2 + F (uA)+ F (uB ) dx.
∂uA ∂t = κ∆µA,µA = 1 2 ( α∆uA + f (uA))(= ∂uA ΨΩ), ∂uB ∂t = κ∆µB ,µB = 1 2 ( α∆uB + f (uB ))(= ∂uB ΨΩ), uA + uB =0,µA + µB =0
µA µB = α∆uA + f (uA). Downloaded 09/12/19 to 128.83.63.20. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
WealsomentionthestochasticCahn–Hilliardequation(alsocalledtheCahn–Hilliard–Cookequation),whichtakesintoaccountthermalfluctuations(see[40,41,44,45,86,89, 145,147,152,153,178,213,262,263,281,446]).
Then,animportantgeneralizationoftheCahn–HilliardequationistheviscousCahn–Hilliardequation,
proposedbyA.Novick-Cohenin[414]toaccountforviscosityeffectsinthephaseseparationofpolymer/polymersystems(seealso[23,88,126,187]).TheviscousCahn–Hilliardequationcanalsobeseenasaparticularcaseofthegeneralizationsproposedby M.Gurtinin[284](which,inparticular,alsoaccountforanisotropy)andwhicharebased onamicroforcebalance,i.e.,anewbalancelawforinteractionsatamicroscopiclevel(see [52,53,55,112,174,175,256,267,369,370,371,386,391,396,403,437,438,439,485] forthemathematicalanalysis);wealsoreferthereadertoyetanotherapproachproposed byP.Podio-Guidugliin[429]andstudiedin,e.g.,[131,132,133,134,139].
AnotherimportantgeneralizationoftheCahn–Hilliardequationisthehyperbolicrelaxationoftheequation,
proposedin[229,230,231,232,334]tomodeltheearlystagesofspinodaldecomposition incertainglasses(seealso[54,244,245,270,271,272,445]forthemathematicalanalysis and[443,444]forthehyperbolicrelaxationoftheCahn–Hilliard–Oonoequationinthe wholespace).Actually,thehyperbolicrelaxationoftheequationisaparticularcaseof moregeneralmemoryrelaxations(foranexponentiallydecreasingmemorykernel),which arestudied,e.g.,in[140,142,144,246,247](seealso[431]).
WealsomentiontheconvectiveCahn–Hilliardequation
whichdescribesthedynamicsofdrivensystems,suchasfacetingofgrowingthermodynamicallyunstablecrystalsurfaces(see[170,171,172,258,346,482]forthemathematical analysis).
Itisimportanttonotethat,inrealisticphysicalsystems,quenchesareusuallycarried outoverafiniteperiodoftime,sothatphaseseparationcanbeginbeforethefinalquenchingisreached.ItisthusimportanttoconsidernonisothermalCahn–Hilliardmodels.Such modelsarederivedandstudiedin[14,15,225,226,394,457].
TheCahn–HilliardequationcanbecoupledwiththeAllen–Cahnequation(see[416]). Thisproblemisstudied,e.g.,in[39,149,348,392,393,416,492,503].
Itcanalsobecoupledwiththeequationsforelasticityorviscoelasticitytoaccountfor mechanicaleffects(see,e.g.,[19,38,46,47,87,157,235,236,237,369,370,421,422, 423,424,436]).
WealsomentionthecouplingoftheCahn–HilliardequationwiththeNavier–Stokes equationsinthecontextoftwo-phase(multiphase)flows(see,e.g.,[2,4,59,61,62,63,64, 66,82,85,111,119,127,208,209,220,221,224,255,285,292,305,320,323,325,332, 347,354,397,499,504])andsomerelatedmodels,suchastheCahn–Hilliard–Hele-Shaw andCahn–Hilliard–Brinkmanequations(see,e.g.,[58,143,154,155,156,201,252,254, 289,480,481,486,498]).Relatedmodelscanalsobeusedtomodeltumorgrowth(see, e.g.,[130,135,136,137,148,168,210,239,240,308,353]).
Chapter1.Introduction11
β∆ ∂u ∂t + ∂u ∂t + ακ∆2 u κ∆f (u)=0,β> 0,
β ∂2u ∂t2 + ∂u ∂t + ακ∆2 u κ∆f (u)=0,β> 0,
∂u ∂t + ακ∆2 u + u ·∇u κ∆f (u)=0,
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Finally,wereferthereaderto,e.g.,[1,10,11,17,18,23,24,26,27,28,29,30,43, 56,63,64,65,66,67,69,70,83,84,94,96,97,100,101,118,120,122,146,158,159, 180,181,182,186,194,197,198,199,200,210,211,212,243,259,264,265,268,282, 283,286,287,288,291,298,299,306,307,314,315,317,319,320,321,322,323,324, 325,335,336,337,338,339,342,343,344,350,356,361,368,406,407,408,434,441, 454,455,461,462,469,477,478,479,483,487,490,491,493,495,496,506,507] forthenumericalanalysisandsimulationsoftheCahn–Hilliardequation(andseveralof itsgeneralizations).Notethat,assuggestedin[182],itisingeneralpreferabletobuild numericalschemesfortheCahn–Hilliardsystem(1.1)ratherthantheequivalentfourthorder-in-spaceCahn–Hilliardequation.Thishastheadvantageofsplittingthefourth-order equationintoasystemoftwosecond-orderequationswhichareeasiertodealwith.Note thatwenowhaveverynicesimulations,alsointhreespacedimensions.
Somewordsonthenotation
Wedenoteby · theusualnormon L2(Ω) and L2(Ω)n (withassociatedscalarproduct ((· , ·))).Moregenerally,asalreadymentioned, · X denotesthenormontheBanach space X;italsodenotesthenormon X n
Throughoutthisbook,letterssuchas c, c , c ,etc.,denoteconstantswhichmaychange fromlinetoline,oreveninthesameline.Similarly,thesameletter Q denotesmonotone increasing(withrespecttoeachargument)functionswhichmaychangefromlinetoline, oreveninthesameline.
Somefinalwords
Wedonotpretend,anddonoteventry,tobeexhaustiveinwhatfollows;itsufficesto typeCahnonMathSciNettounderstandthatthisissimplyimpossible.Forinstance,we willnotdiscussfurtherthenonlocalCahn–Hilliardequation.Itistheauthor’spreference toconcentrateonthelocalCahn–Hilliardequation.Wenotethat,eventhoughitsderivationisphenomenological,itissimpleandrelativelyeasytoimplementnumerically,giving verygoodresults.Thiscanexplainitspopularity(e.g.,amongengineers)anditsusein somanydifferentcontexts.Bycomparison,thenonlocalCahn–Hilliardequationhasa solidphysicalbackgroundbutismuchmoreinvolvedtoimplementnumerically.Actually,thenonlocalCahn–Hilliardequationcertainlydeservesitsownbook,andwedonot underestimateitsimportance.
12Chapter1.Introduction
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Preliminarymaterials
Inthischapter,wegivepreliminarymaterials(linearoperators,compactnessresults,inequalities,globalattractors)whichwillbeusefulinthesucceedingchapters.Inwhat follows, Ω isaboundedandregular(asregularasneeded)domainof Rn , n =1, 2,or 3
2.1 Linearoperators
Werefertheinterestedreaderto[465]forfurtherdevelopmentsonlinearoperators.
Weconsiderthespaces L2(Ω) and H 1(Ω),which,endowedwiththeirusualscalar productsandassociatednorms,areHilbertspaces.
Ofcourse, (u,v) → ((∇u, ∇v)) isnotascalarproducton H 1(Ω),asitisnotcoercive. Toovercomethis,weset u = 1 Vol(Ω) Ω udx,u ∈ L1(Ω), u = 1 Vol(Ω) u, 1 ,u ∈ H 1(Ω), where H 1(Ω) isthetopologicaldualof H 1(Ω), H 1(Ω)= H 1(Ω) ,and ·, · denotes thedualitypairingbetween H 1(Ω) and H 1(Ω).
Wethenset H = ˙ L2(Ω)= {u ∈ L2(Ω), u =0}, V = ˙ H 1(Ω)= H 1(Ω) ∩ H.
ThesespacesarealsoHilbertspaceswhenendowedwiththeinducedscalarproducts. Furthermore, (( , ))V =((∇·, ∇·)) isascalarproducton V ,withassociatednorm · V , whichisequivalenttotheusual H 1(Ω)-norm(owingtothePoincaré–Wirtingerinequality; seebelow).
Let V bethetopologicaldualof V .Then,weknowfromRiesz’srepresentationtheoremthat, ∀l ∈ V ,thereexistsaunique u ∈ V suchthat ((u,v))V = l,v ∀v ∈ V ,where ·, · alsodenotesthedualitypairingbetween V and V
Identifying H withitstopologicaldual H ,wehavetheHilberttriplet V ⊂ H ≡ H ⊂ V ,withdense,continuous,andcompactembeddings.Furthermore,if u and v arein H and V ,respectively,then u,v =((u,v)).Wenotethatwedonotidentify L2(Ω) with
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to
14Chapter2.Preliminarymaterials
itsdualandonlywrite H 1(Ω) ⊂ L2(Ω) and L2(Ω) ⊂ H 1(Ω),withdense,continuous, andcompactembeddings.However,if u ∈ H = H ,thenitcanbeextendedtoalinear andcontinuousformon L2(Ω),withthesamenorm u ,bysetting u,v L2(Ω) ,L2(Ω) = ((u,v)) ∀v ∈ L2(Ω),sothat u ∈ H 1(Ω) and u, 1 H 1(Ω),H 1(Ω) =((u, 1))=0.We canalsoprovethat
V = {u ∈ H 1(Ω), u =0}.
Thischaracterizationisnotstraightforward,however;itfollowsforinstancefromthecharacterizationofthespace H 1(Ω) (see,e.g.,[6]).Alsonotethatif u ∈ V ,thenitfollows fromtheHahn–Banachtheoremthatitcanbeextendedtoalinearandcontinuousformon H 1(Ω) withthesamenorm,sothat u ∈ H 1(Ω)
Remark2.1. Theabovecharacterizationcanalsobeprovedasfollows.Let u ∈ V .Then, thereexistsasequence (uk)k∈N in H and,thus,in V and H 1(Ω) (seeabove)suchthat uk → u in V as k → +∞.Furthermore,for v ∈ H 1(Ω), uk,v H 1(Ω),H 1(Ω) =((uk,v))=((uk,v − v ))= uk,v − v V ,V
Therefore, uk,v H 1(Ω),H 1(Ω) → u,v − v V ,V as k → +∞.Let l : H 1(Ω) → R bedefinedas l(v)= u,v − v V ,V ,v ∈ H 1(Ω).
Wenotethat
|l(v)|≤ u V v − v H 1(Ω) ≤ c v H 1(Ω),v ∈ H 1(Ω),
whichyieldsthat l ∈ H 1(Ω) and uk → l in H 1(Ω) as k → +∞.Notingfinallythat l coincideswith u on V and l, 1 H 1(Ω),H 1(Ω) =0,weindeedhave u =0 (calling u this extension).Conversely,let l ∈{u ∈ H 1(Ω), u =0}.Then,notingthat sup v∈V, v H1(Ω)=1
l(v) ≤ sup v∈H 1(Ω), v H1(Ω)=1 l(v), wededucethat l ∈ V .Furthermore,since l =0,wehave
l(v)= l(v − v )= l,v − v V ,V ∀v ∈ H 1(Ω).
Wecanthendefinethelinearoperator A : V → V as Au,v =((u,v))V ∀u,v ∈ V.
Thisoperatorisanisomorphismfrom V onto V . Wenowset
D(A)= A 1(H)= {u ∈ V,Au ∈ H} = {u ∈ H 1(Ω), ∆u ∈ L2(Ω)}
andcallitthedomainof A.Notethat,indeed,if ((u,v))V =((f,v)) ∀v ∈ V andfor f ∈ H,then
((∇u, ∇v))=((f,v)) ∀v ∈ H 1(Ω)
(itsufficestoreplace v by v − v ).Taking v ∈D(Ω) ≡C∞ c (Ω),itiseasytoseethat ∆u = f inthesenseofdistributionsandthusin L2(Ω). Downloaded 09/11/19 to 128.83.63.20. Redistribution
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subject to
or copyright;
Next,since u ∈ H 1(Ω) and ∆u ∈ L2(Ω),thetrace ∂u ∂ν canbedefinedin H 1 2 (Γ),and ageneralizedformofGreen’sformulaisvalidforevery v ∈ H 1(Ω) (see[466];seealso [465,ChapterII,Example2.5]),yielding
((∆u,v))= ∂u
,v H 1 2 (Γ),H 1 2 (Γ) +((∇u, ∇v)) ∀ v ∈ H 1(Ω).
Wethusdeducethat ∂u ∂ν ,v H 1 2 (Γ),H 1 2 (Γ) =0 ∀ v ∈ H 1(Ω),
sothat
=0onΓ (in H 1 2 (Γ)).Thus,itfollowsfromclassicalellipticregularityresults(see[7,8,9])that u ∈ H 2(Ω).Finally, D(A)= u ∈ H 2(Ω) ∩ V,
Remark2.2. Moregenerally,if f ∈ H m(Ω), m ≥ 0,then u ∈ H m+2(Ω)
2.1.1 Spectralpropertiesoftheoperator A
First,notethat A isself-adjoint(since ((· , ·))V issymmetric).Furthermore,since V ⊂ H iscompact,then A 1 : H → H iscompact(andself-adjoint).Indeed, A 1 : H → D(A) iscontinuous,sothat A 1 : H → V isalsocontinuous(notethatitfollowsfrom theregularitymentionedabovethattheembedding D(A) ⊂ V iscontinuous),andwe concludeowingtothecompactembedding V ⊂ H
Wethusconcludethat A 1 iscompact,self-adjoint,andpositive(asanoperatoron H). Therefore,thereexistsanorthonormalbasis (wj ), j ∈ N,of H formedofeigenvectorsof A 1: A 1 wj = µj wj ,µj → 0as
Since wj = 1 µj A 1wj ∈ D(A),then Awj = λj wj ,λj = 1 µj , and wj , λj areeigenvectors/eigenvaluesof A,where 0 <λ1 ≤ λ2 ≤···, λj → +∞ as j → +∞ (notethat Awj ,wj = λj wj 2 > 0).Furthermore,the wj ’sareorthogonalin V for (( , ))V .Indeed,if j = k,then
((wj ,wk))V = Awj ,wk = λj ((wj ,wk))=0
However,thisfamilyisnotorthonormal,since Awj ,wj =((wj ,wj ))V = λj wj 2 = λj .
2.1.Linearoperators15
∂ν
∂ν
∆u
∂ν
∂u ∂ν
∂u
=0onΓ , and Au = f , u ∈ D(A) and f ∈ H,isequivalentto
= f inΩ, ∂u
=0onΓ
j →
∞
+
,µj > 0
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to
