Table Of ContentTrends in Mathematics
Piotr Drygas´
Sergei Rogosin
Editors
Modern
Problems
in Applied
Analysis
Trends in Mathematics
TrendsinMathematicsisaseriesdevotedtothepublicationofvolumesarisingfrom
conferences and lecture series focusing on a particular topic from any area of
mathematics.Itsaimistomakecurrentdevelopmentsavailabletothecommunityas
rapidlyaspossiblewithoutcompromisetoqualityandtoarchivetheseforreference.
ProposalsforvolumescanbesubmittedusingtheOnlineBookProjectSubmission
Formatourwebsitewww.birkhauser-science.com.
Materialsubmittedforpublicationmustbescreenedandpreparedasfollows:
Allcontributionsshouldundergoareviewingprocesssimilartothatcarriedoutby
journals and be checked for correct use of language which, as a rule, is English.
Articles without proofs, or which do not contain any significantly new results,
shouldberejected.Highqualitysurveypapers,however,arewelcome.
We expecttheorganizerstodelivermanuscriptsina formthatisessentiallyready
fordirectreproduction.AnyversionofTEX isacceptable,buttheentirecollection
of files must be in one particular dialect of TEX and unified according to simple
instructionsavailablefromBirkhäuser.
Furthermore, in order to guarantee the timely appearance of the proceedings it is
essentialthatthefinalversionoftheentirematerialbesubmittednolaterthanone
yearaftertheconference.
Moreinformationaboutthisseriesathttp://www.springer.com/series/4961
Piotr Drygas´ • Sergei Rogosin
Editors
Modern Problems in Applied
Analysis
Editors
PiotrDrygas´ SergeiRogosin
DepartmentofMathematics&Natural DepartmentofEconomics
Sciences BelarusianStateUniversity
RzeszowUniversity Minsk,Belarus
Rzeszów,Poland
ISSN2297-0215 ISSN2297-024X (electronic)
TrendsinMathematics
ISBN978-3-319-72639-7 ISBN978-3-319-72640-3 (eBook)
https://doi.org/10.1007/978-3-319-72640-3
LibraryofCongressControlNumber:2018934721
©SpringerInternationalPublishingAG,partofSpringerNature2018
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisbookispublishedunderthetradenameBirkhäuser,www.birkhauser-science.combytheregistered
companySpringerInternationalPublishingAGpartofSpringerNature.
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
The book includes extended texts of plenary lectures presented at the third BFA
Workshop. The third International Workshop “Boundary Value Problems, Func-
tional Equations and Applications” (BFA3) was held April 20–23, 2016, at the
University of Rzeszów, which is the biggestacademic institution in Podkarpackie
voivodeshipsituatedintheSouth-EastofPoland.Themeetingwasdevotedtorecent
research in the field of differentialand functionalequationsand complexand real
analysis, with a special emphasis on topics related to boundary value problems.
The main attention in the reports at BFA3 was paid to different applications of
the developed technique. The workshop is organized under the guidance of the
InternationalSocietyforAnalysis,itsApplicationsandComputation(ISAAC).The
venueoftheworkshopisrelatedtotwo previousBFA meetingsthattookplacein
Kraków,in2008and2010.
Formally, the content of the book can be divided into two parts. The first
part contains the contributions developing the technique related to the scope of
the workshop, namely differential and functionalequations and real and complex
analysis.
ThepaperbyYu.Alkhutovetal.presentsabriefsurveyofrecentnewresultsfor
thedegeneratepartialdifferentialequationsofp-Laplaciantypeinaboundedcone.
The Dirichlet problem for such equation with the strong nonlinear right part and
theRobinproblemforsuchequationwithsingularnonlinearityintherightpartare
considered.Suchproblemsareappliedtoestablishmathematicalmodelsoccurring
inreaction-diffusiontheory,non-Newtonianfluidtheory,non-Newtonianfiltration,
theturbulentflowofagasinporousmedium,inelectromagneticproblems,inheat
transfer problems, in Fick’s law of diffusion, etc. The aim of the investigation is
to study the behavior of weak solutionsto the problemin the neighborhoodof an
angularorconicalboundarypointoftheboundedcone.
The paper by A.O. Celebi deals with a Dirichlet-type problem, known as the
Riquier problem, the higher order linear complex differential equation in the unit
polydiscofC2.TheconstructedGreen’sfunctionisusedtoobtainthesolutiontoa
modelequationwithhomogeneousboundaryconditions.
v
vi Preface
InthepaperbyM.Dolfinetal.,themodelsbasedontheuseoftheMarkovjump
processesare discussed. It presentsa mathematicaldescriptionat the microscopic
level of the redistribution of individuals in a closed domain featuring, as an
example, an elevator. Starting from the microscopic mathematical representation,
thecorrespondingmodelatthemesoscopiclevelisalsoconsidered.Thepresented
modelrefersto the position of everyseparate individual(an agentof the system).
Two main classes of probabilitiesare introduced:one characterizingthe influence
of the domain,representingthe wall and the entrance of the elevator,and another
onecharacterizingtheeffectoftheinteractionsamongagents.Thefirstoneisdue
to two contributions, respectively describing the tendency to concentrate close to
theentranceandthetendencytostayclosetotheboundary.Thesecondonesimply
representsthetendencytobenottooclosetootheragents.
A methodoffunctionalequationsformixedproblemsfortheLaplaceequation
in multiply connected domains is developed in the paper by V. Mityushev. The
consideredvariantofthemethodisbasedontheequivalentreductionoftheproblem
totheRiemann-Hilbertboundaryvalueproblemforanalyticfunctions.Aproblem
having applications in composites with a discontinuous coefficient on one of the
boundarycomponentsis discussed, providingthat the domain is a canonical one,
namely,lower half-plane with circular holes. A constructiveiterative algorithmto
obtainanapproximatesolutionisproposedintheformofanexpansionintheradius
oftheholes.
A boundary integral equation method is applied to solve the Riemann-Hilbert
probleminthepaperbyM.Nasser.Thesolvabilityoftheboundaryintegralequation
with the generalized Neumann kernel is studied for multiply connected domains
with smooth boundaries. The boundary integral equation with the generalized
Neumann kernel has been used in the study of the Riemann-Hilbert problem in
multiplyconnecteddomainsforspecialclassesoffunctions.Forsuchspecialcases,
the boundary integral equation with the generalized Neumann kernel has been
previously used successfully to compute the conformal mapping onto canonical
domains and to solve several boundary value problems such as the Dirichlet
problem, the Neumann problem, and the mixed boundary value problem. On the
baseofthedevelopedmethod,analternativeprooffortheexistenceanduniqueness
ofthesolutionofthegeneralconjugationproblemisprovided.
The aim ofthe paperby V. Vasilyevis to describenewinterestingexamplesof
nonsmoothmanifoldsandellipticpseudo-differentialoperatorsactinginfunctional
spacesonsuchmanifolds(e.g.,intheSobolev-Slobodetskiitypespaces).Fredholm
properties for these operators are studied by factorization methods in the Vishik-
Eskinframe,andthesearebasedonthepropertiesoffunctionsofseveralcomplex
variables.
ThepaperbyM.Zimaisdevotedtothestudyofafirst-orderdifferentialsystem
subject to a nonlocal condition. The goal in this paper is to establish conditions
sufficientfortheexistenceofpositivesolutionswhentheconsideredproblemisat
resonance. The key tool in this approach is Leggett-Williams norm-type theorem
for coincidences due to O’Regan and Zima. The paper is concluded with several
examplesillustratingthemainresult.
Preface vii
Thesecondpartisdevotedtoapplicationsofsolidandfluidmechanics,aswell
asbiomechanics.The centralpaperof thispartbyI. AndrianovandV. Mityushev
startswiththediscussionofthemeaningoftheterm“exactformula”asitisusedin
thetheoryofcompositematerials.Theeffectivepropertiesofcompositesandreview
literatureonthemethodsofRayleigh,Natanzon–Filshtinsky,functionalequations,
and asymptotic approaches are outlined. In connection with the above methods
and new recent publicationsdevoted to composites, the terms analytical formula,
approximatesolution,closedformsolution,asymptoticformula,etc.frequentlyused
inappliedmathematicsandengineeringinvariouscontextsarediscussed.Though
mathematicians give rigorous definitions of exact form solution the term “exact
solution”continuestobeusedtoolooselyanditsattributesarelost.Inthepresent
paper,examplesofmisleadingusageofsuchatermaregiven.ThepaperbyS.Bock
gives an overview of recently developed spatial generalizations of the celebrated
Kolosov-Muskhelishvili formulae of elasticity theory using the framework of
hypercomplexfunction theory.Based on these results, a hypercomplexversion of
theclassicalKelvinsolutionisobtained.Forthispurposeanewclassofmonogenic
functionswith(logarithmic)linesingularitiesisstudied,andanassociatedtwo-step
recurrence formula is proved. Finally, a connection of the function system to the
Cauchy-kernelfunctionisestablished.
A new analytic model describing the behavior of the viscoelastic periodontal
ligament after the tooth root translational displacement is suggested in the paper
byS.Bosiakovetal.Thesystemofdifferentialequationsfortheplane-strainstate
oftheviscoelasticperiodontalligamentisusedasthegoverningone.Theboundary
conditionscorrespondingtotheinitialsmalldisplacementoftherootandfixedouter
surfaceoftheperiodontalligamentinthedentalalveolusareemployed.Asolutionis
foundnumericallyforthefractionalviscoelasticitymodelassumingthatthestresses
relaxationintheperiodontalligamentafterthecontinuingdisplacementofthetooth
rootoccurswithinaboutfivehours.Theobtainedresultscanbeusedforsimulation
oftheboneremodelingprocessduringorthodontictreatmentandforassessmentof
optimalconditionsoftheorthodonticloadapplication.
The paper by P. Drygas´ is devoted to boundary value problems for elastic
problems modeled by the biharmonic equation in two-dimensionalcomposites. It
is done in the frame of Mityushev’s approach. All the problems are studied via
the method of complex potentials. The considered boundary value problems for
analyticfunctionsarereducedtofunctional–differentialequations.Applicationsto
calculationoftheeffectivepropertiestensorarediscussed.
The boundary value problem of the theory of elasticity for a finite anisotropic
plate with random cracks has been solved in the paper by L. Filshtinskii et al.
Stress intensity factors and energy flows near the tips of cracks are determined
as a linear functional on solutions to a system of singular integral equations.
It is shown that, in case of the normal distribution of the cracks, the statistical
characteristics(mathematicalexpectationsanddispersions)ofthedistraction(stress
intensityfactorsandenergyflows)havealsothenormallawdistribution.
In the paper by S. Gluzman and D. Karpeyev, constructive solutions for the
effectivepropertiesforthreeproblemsinthe field ofrandomstructuredmediaare
viii Preface
presented.Theyareallbasedontruncatedseriesandonaconstructiveinvestigation
of their behavior near divergence points where the physical percolation or phase
transitionsoccur.(1)Effectiveconductivityof2Dconductorswitharbitrarycontrast
parametersis reconstructedfromthe expansionat smallconcentrationsand of the
critical behavior at high concentrations. (2) Effective shear modulus of perfectly
rigid spherical inclusions randomly embedded into an incompressible matrix is
reconstructedgivenitsexpansionatsmallconcentrationsandcriticalbehavior.(3)
AtruncatedFourierexpansiontostudyspontaneousdirectionalorderinginmodels
ofplanarfullyconnectedsuspensionsofactivepolarparticlesisalsoemployed.The
mainresultisthediscoveryofadiscontinuous,abrupttransitionfromanorderedto
adisorderedstate.Itisamacroscopiceffectcausedbyamesoscopicself-quenching
noise.Therelaxationtimeremainsfiniteatthecriticalpoint;thereforetheeffectof
self-quenching is to strongly suppress the critical slowing down and improve the
reactiontimetoexternalstimuli.
An introductory study of gravity-driven Stokesian flow past the wavy bottom,
basedonAdler’setal.papers,isproposedbyR.WojnarandW.Bielski.Inexamples
thewavinessisdescribedbyasinusfunctionanditsamplitudeissmall,uptoO."2/.
AcorrectiontoHagen-Poiseuille’stypefree-flowsolutionisfound.Acontribution
ofcapillarysurfacetensionisdiscussed.
The book is addressed to those who are interested in the developmentand the
realuseoftheinnovativeresearchresults,whichcanhelpsolvescientific andreal
lifeproblems.
Rzeszów,Poland PiotrDrygas´
Minsk,Belarus SergeiRogosin
September2017
Contents
BoundaryValueProblemsfortheSingularp-andp.x/-Laplacian
EquationsinaCone.............................................................. 1
YuryAlkhutov,MikhailBorsuk,andSebastianJankowski
Exactand“Exact”FormulaeintheTheoryofComposites................. 15
IgorAndrianovandVladimirMityushev
On a Hypercomplex Version of the Kelvin Solution
inLinearElasticity............................................................... 35
SebastianBock
ViscoelasticBehaviorofPeriodontalLigament:StressesRelaxation
atTranslationalDisplacementofaToothRoot............................... 51
S.Bosiakov,G.Mikhasev,andS.Rogosin
DirichletTypeProblemsinPolydomains...................................... 65
A.OkayÇelebi
A Microscopic Model of Redistribution of Individuals Inside
an‘Elevator’...................................................................... 77
MarinaDolfin,MirosławLachowicz,andAndreasSchadschneider
New Approach to Mathematical Model of Elastic
inTwo-DimensionalComposites ............................................... 87
PiotrDrygas´
Statistical Characteristics of the Distraction Parameters
in the Unbounded Anisotropic Plane Weakened by Multiple
RandomCracks.................................................................. 101
L.A.Filshtinskii,D.M.Nosov,andYu.V.Shramko
PerturbativeExpansions andCriticalPhenomenain Random
StructuredMedia ................................................................ 117
SimonGluzmanandDmitryA.Karpeyev
ix
