Table Of ContentSpringer INdAM Series 22
Pierluigi Colli
Angelo Favini
Elisabetta Rocca
Giulio Schimperna
Jürgen Sprekels Editors
Solvability,
Regularity, and
Optimal Control
of Boundary Value
Problems for PDEs
In Honour of Prof. Gianni Gilardi
Springer INdAM Series
Volume 22
Editor-in-Chief
G.Patrizio
SeriesEditors
C.Canuto
G.Coletti
G.Gentili
A.Malchiodi
P.Marcellini
E.Mezzetti
G.Moscariello
T.Ruggeri
Moreinformationaboutthisseriesathttp://www.springer.com/series/10283
Pierluigi Colli • Angelo Favini (cid:129) Elisabetta Rocca (cid:129)
Giulio Schimperna (cid:129) JuRrgen Sprekels
Editors
Solvability, Regularity,
and Optimal Control
of Boundary Value Problems
for PDEs
In Honour of Prof. Gianni Gilardi
123
Editors
PierluigiColli AngeloFavini
DipartimentodiMatematica DipartimentodiMatematica
UniversitaJdiPavia UniversitaJdiBologna
Pavia,Italy Bologna,Italy
ElisabettaRocca GiulioSchimperna
DipartimentodiMatematica DipartimentodiMatematica
UniversitàdiPavia UniversitàdiPavia
Pavia,Italy Pavia,Italy
JuRrgenSprekels
Weierstraß-InstitutfürAngewandte
AnalysisundStochastik(WIAS)
Berlin,Germany
DepartmentofMathematics
Humboldt-UniversitätzuBerlin
Berlin,Germany
ISSN2281-518X ISSN2281-5198 (electronic)
SpringerINdAMSeries
ISBN978-3-319-64488-2 ISBN978-3-319-64489-9 (eBook)
DOI10.1007/978-3-319-64489-9
LibraryofCongressControlNumber:2017956717
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Preface
ThisspecialvolumeisdedicatedtoGianniGilardiontheoccasionofhis70thbirth-
day, in tribute to his important achievements in respect of several theoretical and
appliedproblems,especiallyinthefieldsofpartialdifferentialequations,variational
inequalities,optimalcontrol,freeboundaryproblemsandphasetransitionmodels.
GianniGilardiwasborninMilaninFebruary1947.Hestudiedmathematicsat
theUniversityofPavia,wherehegraduatedwithfullmarksinOctober1970.During
thatperiod,hewasalumnusoftheCollegioGhislieri,aprestigioushistoricalcollege
inPaviafoundedbyPopeSt.PiusVin1567.Afterbeingateachingassistantatthe
UniversityofPavia,Giannibecameafullprofessorofmathematicalanalysisatthe
Polytechnic University of Milan in November 1980. He moved back to Pavia in
1985,wherehehasbeenappreciatedasateacheranduniversityprofessorformore
than 30 years. He has taught an impressive number of courses in the Schools of
Engineering,Physicsand Mathematicsat both undergraduateand graduatelevels,
as well as for PhD students. He has been the advisor to a number of master and
PhD students, including some of us editors of the present volume. He has never
spared himself fromhelping colleagues,workingfor the community,or accepting
academic responsibilities. Thus he did not hesitate in accepting the invitation to
serve as chairman of the Department of Mathematics of the University of Pavia,
a position he held for 6 years, or, more recently, as coordinator of the teaching
programsinmathematicsattheUniversityofPavia.Inboththeseroles,hehasbeen
appreciatednotonlybycolleaguesbutalsobytheadministrativestaff.
GiannihasbeenanassociatefellowintheacademyIstitutoLombardoAccademia
di Scienze e Lettere since 2002. He is the author or coauthor of eight books and
of around 100 research papers published in prestigious international journals. He
has given numerous talks in Italy and abroad (Canada, Czech Republic, France,
Germany,Japan,Portugal,Romania,Spain,Switzerland,USA) andcontributedto
theorganizationofalargenumberofconferencesandcourses.
Gianni’s research activity has been intense and varied, being mainly devoted
to the analysis of nonlinear PDEs, but with particular attention to the related
applications. He has primarily been interested in the study of free boundary
v
vi Preface
problemsandphasetransitionmodels.Inmoredetail,wemaymentionamonghis
scientificinterests:
– Well-posedness and regularitytheoryfor second-orderabstract evolutionequa-
tions.
– Monotonicity,speedofpropagationandregularitypropertiesofthefreeboundary
forthe damproblem,the time-dependentdamproblemina generalunbounded
domain,andaregularityresultforthetimederivativeofthesolution.
– Errorestimatesforspace-timediscretizationsofparabolicvariationalinequalities
andaclassofnoncoercivestationaryvariationalinequalities.
– PhasefieldmodelswithmemoryandmoregeneralnonlinearVolterraintegrodif-
ferentialequations.
– Magnetostatic and electrostatic problems in inhomogeneousanisotropic media
with irregularboundaryandmixedboundaryconditions(thisincludesthemost
citedpapercoauthoredbyGianni).
– Phase separation and phase segregation models including also mechanical
effects.
– General phase field systems: Caginalp and Penrose–Fife models, evolutions
basedontheentropybalance,shapememoryalloys,Cahn–Hilliardsystems(also
nonlocal),anddynamicboundaryconditions.
– Diffuseinterfacemodelsdescribingtumorgrowthdynamics.
– Control problems for phase field systems: distributed and boundary optimal
control,slidingmodecontrol,andfeedbackstabilization.
It is a great pleasure for us five editors of this volume to celebrate the 70th
birthdayofourfriendGianni.In additionto beinga teacherto someof us,he has
been a pleasant colleague who could always be approached with questions about
mathematicsortheproofofatechnicallemma,knowingthathewouldbeprepared
to discuss and willing to solve problems. Gianni is very generous in providing
helptoyoungmathematiciansandlessyoungcolleaguesrequiringhisadvicewhen
checkingwhether“thatsolution”couldbeasregularasnecessary.
His webpage contains a number of short notes, lecture notes of courses,
and exercises, with examples and counterexamples that he has generously made
availabletostudentsandcolleagues.Peoplewhohavehadthechancetowritepapers
with him experienced his generosity when, during discussions at the blackboard,
theysomehowbegantoseehowthemathematicalresultswerededuced,withGianni
alreadydeclaringhispersonalwillingnesstowritedownthepaper.
TheappreciationthatGiannialwaysreceivedwithinthescientificcommunityis
reflectedintheenthusiasmwithwhichmanyappliedscientistsandmathematicians
agreed to contribute to this special volume dedicated to him, as announcedin the
beautifulPalazzonediCortonaduringtheINdAMconference“OptimalControlfor
Evolutionary PDEs and Related Topics” in June 2016. We editors of the present
volumearewarmlygratefultoalltheauthorsfortheirpreciouscontributions,which
willsurelybeappreciatedalsobyGianni.
The volume gathers original and peer-reviewed research papers in the field of
partial differential equations, with special emphasis on mathematical models in
Preface vii
phasetransitions,complexfluids,andthermomechanics.Inparticular,thefollowing
thematic areas are developed: nonlinear dynamic and stationary equations, well-
posednessof initial andboundaryvalueproblemsfor systemsof PDEs, regularity
properties for the solutions, optimal control problems and optimality conditions,
and feedback stabilization and stability results. Most of the papers are presented
in a self-contained manner; as a general strategy, the articles describe some new
achievementsand/orthestateoftheartintheirlineofresearch,providinginterested
readerswithanoverviewofrecentprogressandfutureresearchitemsinPDEs.
In conclusion, we would like to join the large family of Gianni, including his
wifeCe,histwodaughtersCarlaandLauraandtheirhusbands,hisfivewonderful
grandchildren,hisfriendsandthecontributorstothepresentvolume,incelebrating
his accomplishments and expressing the wish that he may continue his research
activityformanyyearstocome.LetusconcludewithamottothatGianniwillsurely
appreciate:“Sapientiacumprobitatemorumconiunctahumanæmentisperfectio”.
Pavia,Italy PierluigiColli
Bologna,Italy AngeloFavini
Pavia,Italy ElisabettaRocca
Pavia,Italy GiulioSchimperna
Berlin,Germany JürgenSprekels
July2017
Contents
Rate of Convergence for Eigenfunctions of Aharonov-Bohm
OperatorswithaMovingPole.................................................. 1
LauraAbatangeloandVeronicaFelli
NondecreasingSolutionstoDoublyNonlinearEquations................... 31
GoroAkagiandUlisseStefanelli
IdentificationProblemsforDegenerateIntegro-DifferentialEquations... 55
Mohammed Al Horani, Mauro Fabrizio, Angelo Favini,
andHirokiTanabe
APhaseTransitionModelDescribingAuxeticMaterials.................... 77
ElenaBonetti,MauroFabrizio,andMichelFrémond
Global Well-Posedness for a Phase Transition Model with
IrreversibleEvolutionandAccelerationForces .............................. 97
GiovannaBonfantiandFabioLuterotti
Perimeter Symmetrizationof Some Dynamic and Stationary
EquationsInvolvingtheMonge-AmpèreOperator .......................... 119
BarbaraBrandoliniandJesúsIldefonsoDíaz
OptimalBoundary Controlof a NonstandardCahn–Hilliard
SystemwithDynamicBoundaryConditionandDoubleObstacle
Inclusions ......................................................................... 151
PierluigiColliandJürgenSprekels
NontrivialSolutionsofQuasilinearEllipticEquationswithNatural
GrowthTerm ..................................................................... 183
MarcoDegiovanniandAlessandraPluda
OnaDiffuseInterfaceModelforTumourGrowthwithNon-local
InteractionsandDegenerateMobilities ....................................... 217
SergioFrigeri,KeiFongLam,andElisabettaRocca
ix
x Contents
A Boundary ControlProblemfor the Equationand Dynamic
BoundaryConditionofCahn–HilliardType.................................. 255
TakeshiFukaoandNoriakiYamazaki
NewClassofDoublyNonlinearEvolutionEquationsGoverned
byTime-DependentSubdifferentials........................................... 281
NobuyukiKenmochi,KenShirakawa,andNoriakiYamazaki
BoundednessofSolutionstoaDegenerateDiffusionEquation............. 305
PavelKrejcˇí
OptimalAPrioriErrorEstimatesofParabolicOptimalControl
ProblemswithaMovingPointControl........................................ 327
DmitriyLeykekhmanandBorisVexler
ANoteonthe FeedbackStabilizationofaCahn–HilliardType
SystemwithaSingularLogarithmicPotential ............................... 357
GabrielaMarinoschi
Mathematical Analysis of a Parabolic-Elliptic Model
forBrainLactateKinetics....................................................... 379
AlainMiranville
Weak Formulation for Singular Diffusion
EquationwithDynamicBoundaryCondition ................................ 405
RyotaNakayashikiandKenShirakawa
SmoothandBrokenMinimizersofSomeFreeDiscontinuityProblems... 431
DaniloPercivaleandFrancoTomarelli
Stability Results for Abstract Evolution Equations
withIntermittentTime-DelayFeedback ...................................... 469
CristinaPignotti
From Visco-Energetic to Energetic and Balanced Viscosity
SolutionsofRate-IndependentSystems ....................................... 489
RiccardaRossiandGiuseppeSavaré
ADualityApproachinSomeBoundaryValueProblems.................... 533
DanTiba
OntheStructuralPropertiesofNonlinearFlows ............................ 543
AugustoVisintin
