Table Of ContentSpringerBriefs in Mathematics
Seiro Omata · Karel Svadlenka · Elliott Ginder
Variational Approach
to Hyperbolic Free
Boundary Problems
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Seiro Omata • Karel Svadlenka • Elliott Ginder
Variational Approach
to Hyperbolic Free Boundary
Problems
SeiroOmata KarelSvadlenka
KanazawaUniversity KyotoUniversity
Kanazawa,Japan Kyoto,Japan
ElliottGinder
MeijiUniversity
Tokyo,Japan
ISSN2191-8198 ISSN2191-8201 (electronic)
SpringerBriefsinMathematics
ISBN978-981-19-6730-6 ISBN978-981-19-6731-3 (eBook)
https://doi.org/10.1007/978-981-19-6731-3
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Preface
Thisvolumedealswithfreeboundaryproblemsforpartialdifferentialequationsof
hyperbolic type. The focus is on hyperbolic problems with variational structure.
Then a weak solution can be constructed as a limit of approximate solutions,
whichareinturnobtainedastime-interpolatedminimizersofvariationalfunctionals
with space variableskept continuousand time variable discretized. We attempt to
summarize all that is known to date about this analytical approach. In the second
part of the book, we pursue extensions to more complicated, in particular vector-
valued, problems.For example, we discuss the problem of a rotating elastic body
contactingasolidobstacle,theproblemofabouncingelasticshell,ortheoscillatory
motionbymeancurvatureofanetworkofinterfaceswithjunctions.
Kanazawa,Japan SeiroOmata
Kyoto,Japan KarelSvadlenka
Tokyo,Japan ElliottGinder
July2022
v
Contents
1 Introduction ................................................................... 1
2 PhysicalMotivation........................................................... 3
2.1 MembraneCollisionwithanObstacle ................................. 3
2.2 PeelingofAdhesiveTape ............................................... 5
2.2.1 EnergyatEquilibrium ......................................... 6
2.2.2 Time-DependentProblem ..................................... 7
2.2.3 RegularizedEquationandItsLimit ........................... 9
2.3 DropletMotion:AVolume-PreservingProblem ....................... 13
2.3.1 Volume-ConstrainedProblem Without Free
Boundary ....................................................... 13
2.3.2 Volume-ConstrainedProblemwithFreeBoundary .......... 15
2.3.3 ACoupledModelofMembraneandFluidMotion .......... 17
3 DiscreteMorseFlow.......................................................... 21
3.1 DiscreteMorseFlowfortheLinearWaveEquation ................... 21
3.2 Energy-PreservingSchemefortheLinearWaveEquation ............ 28
3.3 Volume-ConstrainedHyperbolicProblem ............................. 31
4 DiscreteMorseFlowwithFreeBoundary ................................. 39
4.1 WeakSolutionandDiscreteMorseFlow .............................. 39
4.2 ConstructionofApproximateSolutions ................................ 43
4.2.1 Assumptions ................................................... 43
4.2.2 StatementofSemi-discreteProblem .......................... 43
4.2.3 MinimizersAreNonnegative ................................. 44
4.2.4 ExistenceofMinimizers....................................... 44
4.2.5 EnergyEstimate ............................................... 46
4.2.6 SubsolutionProperty .......................................... 47
∞
4.2.7 L -Boundedness .............................................. 50
4.2.8 HölderContinuity ............................................. 52
4.2.9 Euler–LagrangeEquation ..................................... 54
4.2.10 ApproximateWeakSolution .................................. 55
vii
viii Contents
4.3 ExistenceofWeakSolutioninOneDimension ........................ 56
4.3.1 ConvergentSubsequence ...................................... 56
4.3.2 LimitProcess .................................................. 57
5 Energy-PreservingDiscreteMorseFlow................................... 61
5.1 ModifiedDiscreteMorseFlowScheme ................................ 61
5.2 EvolutionofEnergy ..................................................... 62
5.3 ApproximateWeakSolution ............................................ 65
5.4 NumericalResults ....................................................... 72
6 NumericalExamplesandApplications..................................... 77
6.1 FiniteElementApproximation ......................................... 77
6.2 ExamplesofApplications ............................................... 80
6.2.1 Volume-PreservingHyperbolicMeanCurvature
Flow ............................................................ 81
6.2.2 ElastodynamicContactProblem .............................. 84
6.2.3 ElasticShellImpact ........................................... 86
References.......................................................................... 91
Symbols
Ck(D) spaceoffunctionswithcontinuousk-thderivatives
C0,α(D) spaceofHöldercontinuousfunctions
∞
C (D) space of infinitely differentiablefunctions with a compact sup-
0
portinD
C([0,T];V) theBanachspaceofcontinuousmappings[0,T]→V
Hn n-dimensionalHausdorffmeasure
Hk(D) theSobolevspaceWk,2(D)
Hk(D) thesubspaceofHk(D)withzerotraceon∂D
0
(cid:4)·(cid:4) normofSobolevspaceHk(D)
Hk(D)
H1(0,T;V) theSobolev–BochnerspaceW1,2(0,T;V)
Ln n-dimensionalLebesguemeasure
|D| LebesguemeasureofasetD
Lp(D) the Lebesgue space of functions with integrable p-powers for
1≤p <∞
∞
L (D) theLebesguespaceofessentiallyboundedfunctions
p
L (D) theLebesguespaceoffunctionswithlocallyintegrablep-powers
loc
(cid:4)·(cid:4)Lp(D) normofLebesguespaceLp(D)
Lp(0,T;V) the space of Bochner measurable p-integrable functions
(0,T)→V
Wk,p(D) theSobolevspaceoffunctionswithallk-thderivativesinLp(D)
Wk,p(0,T;V) theSobolev–Bochnerspaceoffunctions(0,T)→V
χ thecharacteristicfunctionofasetA
A
∂D theboundaryofasetD
Dη theJacobianmatrixofafunctionη
(cid:6)·,·(cid:7) theinnerproductinL2(D)
{u>0} setofallpointszsuchthatu(z)>0
→ strongconvergenceinanormedlinearspace
(cid:6) weakconvergenceinanormedlinearspace
⇒ uniformconvergence
sptf supportofacontinuousfunctionf
A\B setdifferenceofsetsAandB
D(cid:8) (cid:3)D setD(cid:8) iscompactlyincludedinasetD
ix
Chapter 1
Introduction
This volume is devoted to the study of hyperbolic free boundary problems. Such
problemscan be used to model, among others, the motion of a dropleton a solid
obstacle,themotionofabubbleonawatersurfaceorthevibrationofastringhitting
anobstacle.Forthepurposeofexplanation,werestrictourselvestothescalarcase,
wherethe membranesurroundingthe water or the air in the case of a dropletor a
bubblecanbeexpressedasthegraphofafunction.However,sincethemathematical
analysisoftheinteractionbetweenthemembraneandtheobstaclecanbeextended
tothevector-valuedcase,weshalllateronpresentspecificexamplesinvestigating
suchproblems.
In general,the membraneformsa positive contactanglewith the obstacle, and
therefore the Laplacian or other differential operator describing the shape of the
membraneis onlya measuresupportedin theboundaryofthe contactset, usually
called a free boundary.We will show how to derive mathematicalproblemsfor a
few physicalsystems starting froman action functional.Then we will discuss the
mathematicaltheory,andintroduceapproximationmethodsforobtainingnumerical
solutionsofthemodelequations.
SuchproblemswerefirstinvestigatedbyTa-TsienLiandhiscollaborators(see,
e.g., [43–46]), where a local, and later global, existence theory for a class of
one-dimensional quasilinear hyperbolic free boundary problems was established.
Althoughthestudywassuccessfulinobtainingrathergeneralexistenceresults,we
remarkthatthemotivationoftheirresearchistounderstandproblemssuchasshock
behavior in the generalized Riemann problem for isentropic flow. This viewpoint
specifiestheproblemsettingandtheunderlyingassumptionforwhichtheirtheory
isapplicable.
The current study focuses on a different class of hyperbolic free boundary
problems:thosewhicharecharacterizedbythepresenceofavariationalstructure.
By discretizing time, our hyperbolicproblems can then be recast as sequences of
minimizationproblems,andthisenablesustoemploymoderntechniquesfromthe
calculusofvariations.An advantageof this approachis thatits frameworkcan be
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