Research in PDEs and Related Fields: The 2019 Spring School, Sidi Bel Abbès, Algeria

192 Pages·2022·2.033 MB·English

by  Kaïs Ammari

Tutorials, Schools, and Workshops in the
Mathematical Sciences
Kaïs Ammari
Editor
Research in
PDEs and
Related
Fields
The 2019 Spring School, Sidi Bel
Abbès, Algeria
Tutorials, Schools, and Workshops
in the Mathematical Sciences
Thisserieswillserveasaresourceforthepublicationofresultsanddevelopments
presentedatsummerorwinterschools,workshops,tutorials,andseminars.Written
in an accessible style, they present important and emerging topics in scientific
researchforPhDstudentsandresearchers.Fillingagapbetweentraditionallecture
notes,proceedings,andstandardtextbooks,thetitlesincludedinthisseriespresent
materialfromtheforefrontofresearch.
Manuscripts are solicited by the editorial boards of each volume and then
reviewed by a minimum of three peer reviewers to ensure the highest standards
ofscientificliterature.
Kaïs Ammari
Editor
Research in PDEs and
Related Fields
The 2019 Spring School, Sidi Bel Abbès,
Algeria
Editor
KaïsAmmari
DepartmentofMathematics
UniversityofMonastir
Monastir,Tunisia
ISSN2522-0969 ISSN2522-0977 (electronic)
Tutorials,Schools,andWorkshopsintheMathematicalSciences
ISBN978-3-031-14267-3 ISBN978-3-031-14268-0 (eBook)
https://doi.org/10.1007/978-3-031-14268-0
MathematicsSubjectClassification:35R30,35Q41,35K65,35P20,35R02
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Preface
Thisvolumeconstitutestheproceedingsofthespringschool:“TrendsinPDE’sand
RelatedFields”.
ThisspringschoolwasheldattheUniversityofSidiBelAbbès,Algeria,inthe
period08–10April2019(seehttp://conf-sidi-bel-abbes.ur-acedp.org/fordetails).
The spring school consisted of two mini-courses, seven invited talks on the
theme, and some short talks. This volume gathered the mini-course and the five
contributions.
Theinvitedspeakersagreedtowritereviewpapersrelatedtotheircontributions
to the spring school, while others have written more traditional research papers,
whichconstitutethelastpartofthisvolume.Theyrepresentrecentandnewworks
onthetopicofmathematicalcontroltheoryandrelatedfields.
Webelievethatthisvolumethereforeprovidesanaccessiblesummaryofawide
rangeofactiveresearchtopics,alongwithsomeexcitingnewresults,andwehope
that it will prove a useful resource for both graduate students new to the area and
moreestablishedresearchers.
Thespringschoolbroughttogetherinternationallyleadingresearchersandyoung
researcherswhocamefromallaroundtheworld.Theorganizers’intentionwasto
provideawideanglesnapshotofthisexcitingandfast-movingareaandfacilitatethe
exchangeofideasonrecentadvancesinitsvariousaspects.Thenumerousformal,
informal,andsometimeslivelydiscussionsthatresultedfromthisinteractionwere
forusasignthatweachievedsomethinginthedirectionoffulfillingthisaim.
Our second aim was to ensure that the diffusion of these recent results was
not limited to established researchers in the area and those present at the spring
school but also available to newcomers and more junior members of the research
community. This was reflected by the presence of many unfamiliar and/or young
faces in the audience. The present proceedings should hopefully complete the
fulfillmentofoursecondaim.
Thisspringschoolwouldnothavematerializedwithoutthehelpandsupportof
thefollowinginstitutions.
We are very grateful to Laboratoire d’Analyse et de Contrôle des EDP at the
UniversityofSidiBelAbbèsandtotheResearchLabACPDE,AnalysisandControl
v
vi Preface
of Partial Differential Equations, at the University of Monastir for their financial
supportswithoutwhomthisspringschoolwouldnotbeaccessiblewithoutfees.
We would also like to thank all the participants of the spring school who have
madethiseventasuccess,thecontributorstotheseproceedings.
Monastir,Tunisia KaïsAmmari
May2022
Contents
SobolevSpacesandEllipticBoundaryValueProblems ..................... 1
ChérifAmrouche
SurveyontheDecayoftheLocalEnergyfortheSolutionsofthe
NonlinearWaveEquation....................................................... 35
AhmedBchatnia
ASpectralNumericalMethodtoApproximatetheBoundary
ControllabilityoftheWaveEquationwithVariableCoefficients........... 103
CarlosCastro
AggregationEquationandCollapsetoSingularMeasure................... 123
TaoufikHmidiandDongLi
GeometricControlofEigenfunctionsofSchrödingerOperators........... 151
FabricioMacià
StabilityofaGraphofStringswithLocalKelvin–VoigtDamping......... 169
KaïsAmmari,ZhuangyiLiu,andFarhatShel
vii
Sobolev Spaces and Elliptic Boundary
Value Problems
ChérifAmrouche
2010MathematicsSubjectClassification 35L05,34K35
1 SobolevSpaces, Inequalities,Dirichlet,and Neumann
Problemsforthe Laplacian
1.1 SobolevSpaces
LetusintroducethefollowingSobolevspaces:forany1<p <∞
(cid:2) (cid:3)
Wm,p((cid:2))= u∈D(cid:4)((cid:2)); ∀|α|≤m, Dαu∈Lp((cid:2))
and
(cid:4) (cid:5) (cid:5) (cid:6)
|Dαu(x)−Dαu(y)|p
Ws,p((cid:2))= u∈Wm,p((cid:2)); <∞, ∀|α|=m ,
|x−y|N+σp
(cid:2) (cid:2)
wherem ∈ N,s = m+σ,0 < σ < 1and(cid:2)isanopensetofRN.Equippedwith
thegraphnorm,theyareBanachspaces.
When(cid:2)=RN,usingtheFouriertransform,wedefineforanyrealnumbersthe
space
(cid:4) (cid:5) (cid:6)
Hs(RN)= u∈S(cid:4)(RN); (1+|ξ|2)s|uˆ(ξ)|2dξ <∞ ,
RN
whichisanHilbertspaceforthenorm:
C.Amrouche((cid:2))
LaboratoiredeMathématiquesetLeursApplications,UMRCNRS5142,UniversitédePauetdes
Paysdel’Adour,Pau,France
e-mail:[email protected]
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 1
K.Ammari(ed.),ResearchinPDEsandRelatedFields,Tutorials,Schools,and
WorkshopsintheMathematicalSciences,
https://doi.org/10.1007/978-3-031-14268-0_1
2 C.Amrouche
(cid:7)(cid:5) (cid:8)
1/2
(cid:7)u(cid:7)Hs(RN) = (1+|ξ|2)s|uˆ|2dx .
RN
By Plancherel’s theorem we prove that Ws,2(RN) = Hs(RN) for all s ≥ 0 and
thisidentityisalgebraicalandtopological.So,inthecasep = 2,wedenotemore
simplythespaceWs,2((cid:2))byHs((cid:2)).
Definition1.1 Fors >0and1≤p <∞,wedenote
Ws,p((cid:2))=D((cid:2))(cid:7)·(cid:7)Ws,p((cid:2)),
0
anditstopologicaldualspace
(cid:9) (cid:10)
W−s,p(cid:4)((cid:2))= Ws,p((cid:2)) (cid:4),
0
wherep(cid:4) istheconjugateofp:1/p+1/p(cid:4) = 1.Forp = 2,wewillwriteHs((cid:2))
0
andH−s((cid:2)),respectively.
Proposition1.2 SupposeT ∈ D(cid:4)((cid:2)).ThenT ∈ W−m,p(cid:4)((cid:2)),withm ∈ N∗,ifand
onlyif
(cid:11)
T = Dαf , with f ∈Lp(cid:4)((cid:2)).
α α
|α|≤m
1.2 FirstProperties
It will be assumed from now on that (cid:2) is a bounded open subset of RN with a
Lipschitzboundary.
Letusconsiderthefollowingspace
(cid:12) (cid:13)
D((cid:2))= v|(cid:2); v ∈D(RN) .
Theorem1.3
(i) ThespaceD((cid:2))isdenseinWs,p((cid:2))foranys >0 (evenif(cid:2)isunbounded).
(ii) ThespaceD(RN)isdenseinWs,p(RN)foranys ∈R.
Asconsequence,wehavethefollowingproperty:foranys >0
(cid:14) (cid:15)
(cid:4)
Ws,p(RN)=Ws,p(RN) and W−s,p(cid:4)(RN)= Ws,p(RN) .
0
Butingeneral,foranys >0,wehaveWs,p((cid:2))(cid:2)Ws,p((cid:2)).
0