Partial Differential Equations in Action: Complements and Exercises

433 Pages·2015·3.07 MB·English

by  Sandro Salsa

Sandro Salsa
Gianmaria Verzini
Partial Differential
Equations in Action
Complements and Exercises
TT
XX
EE
TT
II
NN
123
UU
UNITEXT – La Matematica per il 3+2
Volume 87
Moreinformationaboutthisseriesat
http://www.springer.com/series/5418
(cid:2)
Sandro Salsa Gianmaria Verzini
Partial Differential
Equations in Action
Complements and Exercises
SandroSalsa GianmariaVerzini
DipartimentodiMatematica DipartimentodiMatematica
PolitecnicodiMilano PolitecnicodiMilano
Milano,Italy Milano,Italy
TranslatedbySimonG.Chiossi,UFBA–UniversidadeFederaldaBahia,Salvador(Brazil).
TranslationfromtheItalianlanguageedition:Equazioniaderivateparziali.Complementiedesercizi,
SandroSalsaeGianmariaVerzini,©Springer-VerlagItalia,Milano2005.Allrightsreserved.
UNITEXT–LaMatematicaperil3+2
ISBN978-3-319-15415-2 ISBN978-3-319-15416-9(eBook)
DOI10.1007/978-3-319-15416-9
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2015930285
©SpringerInternationalPublishingSwitzerland2015
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartofthe
materialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broad-
casting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformationstorage
andretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknown
orhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsorthe
editorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforanyerrorsor
omissionsthatmayhavebeenmade.
CoverDesign:SimonaColombo,GiochidiGrafica,Milano,Italy
TypesettingwithLATEX:PTP-Berlin,ProtagoTEX-ProductionGmbH,Germany(www.ptp-berlin.eu)
SpringerisapartofSpringerScience+BusinessMedia(www.springer.com)
Preface
Thisbookisdesignedforadvancedundergraduatestudentsfromvariousdisciplines,in-
cludingappliedmathematics,physics,andengineering.ItevolvedduringthePDEcourses
thatbothauthorshavetaughtduringrecentdecadesatthePolitecnicodiMilano,andcon-
sistsofproblemsofvarioustypesanddifficulties.
Inthefirstpartofthebook,whilemuchemphasisisplacedonthemostcommonmeth-
odsofresolution,suchasseparationofvariablesorthemethodofcharacteristics,wealso
invitethestudenttohandlethebasictheoreticaltoolsandpropertiesofthesolutionstothe
fundamentalequationsofmathematicalphysics.
The second part is slightly more advanced and requires basic tools from functional
analysis. A small number of exercises aims to familiarize the student with the first ele-
mentsofthetheoryofdistributionsandoftheHilbertianSobolevspaces.Thefocusthen
switchestothevariationalformulationofthemostcommonboundaryvalueproblemsfor
uniformlyellipticequations.Asubstantialnumberofproblemsisdevotedtotheuseofthe
Riesz representation and the Lax-Milgram theorems together with Fredholm alternative
toanalysewellposednessorsolvabilityofthoseproblems.Next,anumberofproblems
addresses theanalysis ofweaksolutionstoinitial-boundary valueproblemsfortheheat
orthewaveequation.
Thetextiscompletedbytwoshortappendixes,thefirstdealingwithSturm-Liouville
problemsandBesselfunctionsandthesecondlistingfrequentlyusedformulas.
Eachchapterbeginswithabriefreviewofthemaintheoreticalconceptsandtoolsthat
constitutenecessaryprerequisitesforaproperunderstanding.ThetextPartialDifferential
EquationinAction[18],byS.Salsa,isthenaturaltheoreticalreference.
Within each chapter, the problems are divided into two sections. In the first one we
present detailed solutions and comments to provide the student with a reasonably com-
pleteguide.Inthesecondsection,weproposeasetofproblemsthateachstudentshould
trytosolvebyhim-orherself.Ineachcase,asolutioncanbefoundattheendofthechap-
ter.Someproblemsareproposedastheoreticalcomplementsandmayproveparticularly
challenging;thisisespeciallytrueofthosemarkedwithoneortwoasterisks.
Milano,January2015 SandroSalsa
GianmariaVerzini
Contents
1 Diffusion ........................................................... 1
1.1 Backgrounds ................................................... 1
1.2 SolvedProblems ................................................ 3
1.2.1 Themethodofseparationofvariables........................ 3
1.2.2 Useofthemaximumprinciple .............................. 20
1.2.3 Applyingthenotionoffundamentalsolution .................. 25
1.2.4 UseofFourierandLaplacetransforms ....................... 37
1.2.5 Problemsindimensionhigherthanone....................... 43
1.3 FurtherExercises................................................ 50
1.3.1 Solutions ................................................ 56
2 TheLaplaceEquation ............................................... 81
2.1 Backgrounds ................................................... 81
2.2 SolvedProblems ................................................ 84
2.2.1 Generalpropertiesofharmonicfunctions ..................... 84
2.2.2 Boundary-valueproblems.Solutionmethods .................. 95
2.2.3 PotentialsandGreenfunctions ..............................117
2.3 FurtherExercises................................................124
2.3.1 Solutions ................................................130
3 FirstOrderEquations ...............................................149
3.1 Backgrounds ...................................................149
3.2 SolvedProblems ................................................152
3.2.1 Conservationlawsandapplications ..........................152
3.2.2 Characteristicsforlinearandquasilinearequations .............181
3.3 FurtherExercises................................................194
3.3.1 Solutions ................................................197
4 Waves .............................................................215
4.1 Backgrounds ...................................................215
4.2 SolvedProblems ................................................217
viii Contents
4.2.1 One-dimensionalwavesandvibrations .......................217
4.2.2 Canonicalforms.CauchyandGoursatproblems ...............238
4.2.3 Higher-dimensionalproblems...............................247
4.3 FurtherExercises................................................255
4.3.1 Solutions ................................................259
5 FunctionalAnalysis .................................................273
5.1 Backgrounds ...................................................273
5.2 SolvedProblems ................................................278
5.2.1 Hilbertspaces ............................................278
5.2.2 Distributions .............................................291
5.2.3 Sobolevspaces ...........................................298
5.3 FurtherExercises................................................310
5.3.1 Solutions ................................................314
6 VariationalFormulations ............................................333
6.1 Backgrounds ...................................................333
6.2 SolvedProblems ................................................336
6.2.1 One-dimensionalproblems .................................336
6.2.2 Ellipticproblems .........................................346
6.2.3 Evolutionproblems .......................................366
6.3 FurtherExercises................................................381
6.3.1 Solutions ................................................385
AppendixA.Sturm-Liouville,LegendreandBesselEquations................405
A.1 Sturm-LiouvilleEquations........................................405
A.1.1 Regularequations.........................................405
A.1.2 Legendre’sequation.......................................406
A.2 Bessel’sEquationandFunctions...................................407
A.2.1 Besselfunctions ..........................................407
A.2.2 Bessel’sequation .........................................410
AppendixB.Identities ...................................................413
B.1 Gradient,Divergence,Curl,Laplacian ..............................413
B.2 Formulas ......................................................415
B.3 FourierTransforms ..............................................416
B.4 LaplaceTransforms..............................................417
References..............................................................419
1
Diffusion
1.1 Backgrounds
Weshallrecallafewnotionsandresultsthatcropupfrequentlyconcerningthediffusion
equation
u (cid:3)D(cid:2)uDf
t
definedonacylindricaldomainQ D(cid:3)(cid:4).0;T/,where(cid:3)isadomain(connected,open
T
subset)ofRn.Hereu D u.x;t/,andtheLaplacian(cid:2)istakenwithrespecttothespatial
variablesxonly.
•Parabolicboundary.TheunionofthebaseofQ (i.e.(cid:3)(cid:4)¹0º)andthelateralsur-
T
faceS D@(cid:3)(cid:4)Œ0;T(cid:4)istheparabolicboundaryof Q ,whichwedenoteby@ Q .In
T T p T
well-posedproblemsforthediffusionequationthisiswhereonemustassignthedata.
(cid:2) (cid:3)
•Maximumprinciples.Let(cid:3)beboundedandw 2C2;1.Q /\C Q asub-solution
T T
(orsuper-solution),thatis
w (cid:3)D(cid:2)w Dq (cid:5)0 (resp. (cid:6)0) inQ :
t T
Then w reaches its maximum value (resp. minimum) on the parabolic boundary @ Q
p T
ofQ :
T
maxw D max w
QT @pQT
(weakmaximumprinciple).Inparticular,ifw isnegative(resp.positive)on@ Q ,then
p T
itisnegative(resp.positive)overallofQ .If,further,w.x ;t /Dmax withx 2(cid:3);
thenwisconstanton(cid:3)(cid:4)Œ0;t (cid:4)(strongmTaximumprincipl0e).0 QT 0
0
•FundamentalsolutionandglobalCauchyproblem.Thefunction
(cid:5) .x;t/D 1 e(cid:2)jxj2=.4Dt/; t >0;
D .4(cid:6)Dt/n=2
is called fundamental solution to the diffusion equation; when t > 0 it solves
©SpringerInternationalPublishingSwitzerland2015
S.Salsa,G.Verzini,PartialDifferentialEquationsinAction.ComplementsandExercises,
UNITEXT–LaMatematicaperil3+287,DOI10.1007/978-3-319-15416-9_1
2 1 Diffusion
u (cid:3)D(cid:2)uD0andistheuniquefunctionsatisfying
t
Z
lim(cid:5) .x;t/Dı .x/; (cid:5) .x;t/dxD1foranyt >0;
D n D
t#0 Rn
whereı .x/denotesthen-dimensionalDirac’sdeltafunction.
n
ThefundamentalsolutionenablestoconstructthegeneralsolutiontotheglobalCauchy
problem ´
u (cid:3)D(cid:2)uDf .x;t/ inRn(cid:4).0;1/
t
u.x;0/Dg.x/ inRn;
bymeansoftheformula
Z Z Z
t
u.x;t/D (cid:5) .x(cid:3)y;t/g.y/ dyC (cid:5).x(cid:3)y;t (cid:3)s/f .y;s/ dyds:
D
Rn 0 Rn
Thelatterholds,forexample,whenjg.x/j(cid:5)ceAjxj,f isboundedandf,f ,f ,f
t xj xixj
arecontinuousonRn(cid:4).0;C1/.Atapointx ofcontinuityofgwehave
0
u.x;t/!g.x / as .y;t/!.x ;0/,t >0:
0 0
•Random walkand fundamental solution (n D 1).Letusconsideraparticleofunit
massmovingalongthex-axisasfollows.
i) Inatimeinterval(cid:7) theparticleadvancesbyh,startingfromx D0.
ii) Itmovesleftwardsorrightwardswithprobabilityp D 1=2,eachtimeindependently
ofthepreviousaction.
At time t D N(cid:7), i.e. after N iterations, the particle will have reached point x D mh,
whereN isanaturalnumberandmaninteger.Theprobabilityp.x;t/thatitwillbeinx
attimet isthesolutionofthediscreteproblem
1 1
p.x;t C(cid:7)/D p.x(cid:3)h;t/C p.xCh;t/; (1.1)
2 2
withinitialconditions
p.0;0/D1andp.x;0/D0 ifx ¤0:
Passing to the limit in (1.1) for h;(cid:7) ! 0, whilst keeping h2=(cid:7) D 2D D constant and
interpreting p as a probability density, gives the equation p D Dp , and the initial
t xx
conditionsnowread
limp.x;t/Dı .ı Dı/:
1
t#0
Wehavealreadynoticedthattheuniquesolutionwithunitmassisthefundamentalsolu-
tiontothediffusionequation:
p.x;t/D(cid:5) .x;t/:
D