Multi-layer potentials and boundary problems : for higher-order elliptic systems in Lipschitz domains

430 Pages·2013·2.388 MB·English

by  Irina Mitrea, Marius Mitrea

Lecture Notes in Mathematics 2063
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
•
Irina Mitrea Marius Mitrea
(cid:2)
Multi-Layer Potentials
and Boundary Problems
for Higher-Order Elliptic Systems in Lipschitz
Domains
123
IrinaMitrea MariusMitrea
TempleUniversity UniversityofMissouri
DepartmentofMathematics DepartmentofMathematics
Philadelphia,PA,USA Columbia,MO,USA
ISBN978-3-642-32665-3 ISBN978-3-642-32666-0(eBook)
DOI10.1007/978-3-642-32666-0
SpringerHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2012951623
MathematicsSubjectClassification(2010):35C15,35G45,35J48,35J58,35J08,35B30,31B10,31B30
(cid:2)c Springer-VerlagBerlinHeidelberg2013
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Dedicata˘ cu multdragfamilieinoastre
•
Preface
Layer potentials constitute one of the most powerful tools in the treatment of
boundaryvalueproblemsassociatedwith(ellipticandparabolic)partialdifferential
equations(PDEs).Theyhavebeentraditionallyemployedinthecontextofsecond-
orderPDE,andoneofthemaingoalsofthismonographistosystematicallydevelop
a multilayer theory applicable to the higher-order setting. This extension of the
classical theory is carried out in the context of arbitrary Lipschitz domains and
includesmappingpropertiesof such multilayersassociated with complex,matrix-
valued, constant coefficient, homogeneous elliptic systems, in function spaces
suitably adapted to the higher-regularity case (of Besov and Triebel–Lizorkin
type), Carleson measure estimates, non-tangential maximal function estimates,
jump-relations, etc., which turn out to be just as versatile and effective as their
second-ordercounterparts.Inparticular,thistheoryappliestosuchbasicdifferential
operatorslike the Laplacian, the bi-Laplacian,the polyharmonicoperator,and the
Lame´ system of elasticity, thoughthe gist of the presentwork is constructing,for
the first time, a comprehensive theory (of Caldero´n–Zygmund type) for singular
integraloperatorsofmultilayertypeassociatedwithgenerichigher-orderPDEsand
todiscusssomeoftheimplicationsofthismultilayertheorytothewell-posednessof
boundaryvalueproblemsforhigher-orderPDEs.Assuch,oneofthemainpurposes
ofthismonographistoaddressanobviousgap/discrepancy/imbalanceinthepresent
literaturebetweenthesecond-andhigher-ordercase.
The intendedaudienceconsists of any mathematicallytrained scientist with an
interest in boundary value problems and partial differential equations. While this
is an original research monograph, significant effort has been put in to make the
materialasreasonablyaccessibleaspossible.Inparticular,thismonographshould
alsobeusefultojuniorscientistsworkingintheareaofPDE.
Philadelphia,PA IrinaMitrea
Columbia,MO MariusMitrea
June2012
vii
•
Contents
1 Introduction .................................................................. 1
2 Smoothness Scales and Caldero´n–Zygmund Theory
intheScalar-ValuedCase................................................... 21
2.1 LipschitzDomainsandNon-tangentialMaximalFunction........... 21
2.2 SobolevSpacesonLipschitzBoundaries.............................. 31
2.3 BriefReviewofSmoothnessSpacesinRn ............................ 42
2.4 SmoothnessSpacesinLipschitzDomains............................. 48
2.5 WeightedSobolevSpacesinLipschitzDomains...................... 61
2.6 Stein’sExtensionOperatoronWeightedSobolevSpaces
inLipschitzDomains ................................................... 73
2.7 OtherSmoothnessSpacesonLipschitzBoundaries................... 88
2.8 Caldero´n–ZygmundTheoryintheScalar-ValuedCase............... 108
3 FunctionSpacesofWhitneyArrays....................................... 125
3.1 Whitney–LebesgueandWhitney–SobolevSpaces.................... 125
3.2 Whitney–BesovSpaces................................................. 138
3.3 Multi-TraceTheory..................................................... 142
3.4 Whitney–HardyandWhitney–Triebel–LizorkinSpaces.............. 169
3.5 Interpolation............................................................. 180
3.6 Whitney–BMOandWhitney–VMOSpaces........................... 191
4 TheDoubleMulti-LayerPotentialOperator ............................. 199
4.1 DifferentialOperatorsandFundamentalSolutions.................... 199
4.2 TheDefinitionofDoubleMulti-LayerandNon-tangential
MaximalEstimates...................................................... 207
4.3 CarlesonMeasureEstimates............................................ 220
4.4 JumpRelations.......................................................... 233
4.5 EstimatesonBesov,Triebel–Lizorkin,andWeighted
SobolevSpaces.......................................................... 245
ix