Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

446 Pages·2015·4.673 MB·English

by  Sabir Umarov (auth.)

Developments in Mathematics
Sabir Umarov
Introduction to
Fractional and Pseudo-
Differential Equations
with Singular Symbols
Developments in Mathematics
VOLUME41
SeriesEditors:
KrishnaswamiAlladi,UniversityofFlorida,Gainesville,FL,USA
HershelM.Farkas,HebrewUniversityofJerusalem,Jerusalem,Israel
Moreinformationaboutthisseriesathttp://www.springer.com/series/5834

Sabir Umarov
Introduction to Fractional
and Pseudo-Differential
Equations with Singular
Symbols
123
SabirUmarov
DepartmentofMathematics
UniversityofNewHaven
WestHaven,CT,USA
ISSN1389-2177 ISSN2197-795X (electronic)
DevelopmentsinMathematics
ISBN978-3-319-20770-4 ISBN978-3-319-20771-1 (eBook)
DOI10.1007/978-3-319-20771-1
LibraryofCongressControlNumber:2015945587
Mathematics Subject Classification (2010): 46Bxx, 46Sxx, 35Sxx, 35Gxx, 32W25, 47G30, 26A33,
60H10,60G50,35Q84
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Preface
The20thcenturywasrichwithgreatscientificandmathematicaldiscoveries.One
ofthemostinfluentialeventsinmathematicswastheintroductionoftheLebesgue
integral(Lebesgue,1901)followedsoonafter by Borel’sdevelopmentof measure
theory(Borel,1903).CombinedwithCantor’ssettheory,whoseaxiomaticformap-
pearedinthebeginningofthecentury(Zermelo,1907),thesesourceswiththeirdeep
andremarkableideasmadepossiblethedevelopmentofalargenumberoffunction
spacesthatareextremelyimportantinmodernanalysis,suchasL -spaces(F.Riesz,
p
1910), Sobolev spaces (Sobolev,1938),Schwatz’s distributions(Schwartz, 1951),
andmanymore.Undoubtedly,modernfunctionalanalysis,includingmeasurethe-
ory,thetheoryoftopologicalvectorspaces,andoperatortheory,withtheirvarious
powerfultechniquesandmethods,ultimatelytakestheiroriginfromtheseremark-
ablesources.
Another extremely influential series of events was the emergenceof the math-
ematical formalismof probabilitytheory(Kolmogorov,1933)in conjunctionwith
theintroductionofBrownianmotion(Einstein,1905;Wiener,1927)andLe´vypro-
cesses(Le´vy,1938).Theseledtoanumberofdeepdevelopmentsinthetheoryof
stochastic processes,two ofwhicharetheItoˆ’sstochastic calculus(Itoˆ,1948)and
the theory of stochastic ordinary and partial differential equations. An important
attribute in this chain is the Fokker-Planck-Kolmogorov equation (Fokker, 1913;
Planck,1917;Kolmogorov,1931),whichprovidesa deterministicwayofdescrib-
ingstochasticprocesses.
By the1950sthe ideasbehindSobolevspaces werewell understood,Schwartz
distributionswere introduced(justifying,in particular,theDirac deltafunctionin-
tensively used in physics), mathematical theories of Brownian motion and Le´vy
processesweredeveloped,andmathematicalreasoningofstochasticintegrals,Itoˆ’s
stochastic calculus had emerged (justifying the Langevin stochastic differential
equationsintroducedin1908).Thisspawnedtherapiddevelopmentinthe1960snot
onlyofnewpowerfulmethodsandgeneralizationsoftheabovetheories,butalsothe
initiationofanumberofnewtheories.Justtomentionafew,thesearethetheoryof
pseudo-differentialoperators(J.J.KohnandL.Nirenberg1965,andL.Ho¨rmander,
vii
viii Preface
1965)initsmodernform,Fourierintegraloperators(Ho¨rmander,1968–71),contin-
uoustimerandomwalks(MontrollandWeiss, 1965),andfractionalorderFokker-
Planck and diffusion equations. There was an abundance of applications of these
theories in the natural and social sciences and engineering,including the filtering
theory(Kalman,Bucy,1961),etc.Inthescopeofthesetheories,anumberoflong-
standingmathematicalproblemshavefoundtheirsolutions.Therelatedtheoriesof
fractionalorderdifferentialoperatorsandequationshavealsoledtoahugenumber
ofapplicationsinscienceandengineering,aswellasinsidemathematicsitself.
Manygoodbooksandpapershavebeenwrittenabouttheclassicaltheoriesmen-
tioned above, and continue to appear, since extensive investigations, discovering
newconceptsanddevelopingnewtheories,areongoing.Withnopretenseofcom-
pleteness, we referthe reader to the books[Nik77, BIN75, Tri77, Tri83] on func-
tionspaces,[Hor83,Tre80,Tay81,Shu78,Won99]onpseudo-differentialoperators
andFourierintegraloperators,[IW81,Pro91,KSh91,Sit05,App09]onItoˆ calculus
and stochastic differentialequations,and[OS74, SKM87, Rub96, Pod99, KST06,
Mai10]onfractionalcalculusandfractionalorderdifferentialequations.
What is the present book about? This book is an introduction to the theory
of pseudo-differential operators with symbols singular with respect to dual vari-
ables combined with fractional order differential equations and their applications
to variousapplied fields. One of the essential requirementsin the classical theory
of pseudo-differential operators is the symbol must be smooth on the cotangent
bundle, that is a(x,ξ)∈C∞(Ω,Rn). However, in the solution of many problems
ofmathematicalphysicspseudo-differentialoperatorswithsingularsymbolsarise.
Forinstance,solutionoperatorsoftwo-ormulti-pointboundaryvalueproblemsfre-
quentlyappeartohavesymbolssingularindualvariables.Forexample,thesymbol
ofthesolutionoperatorofthefollowingsimplenonlocalboundaryvalueproblem
∂2u(t,x) ∂2u(t,x)
+ (t,x)=0, t∈(0,2), x∈R,
∂t2 ∂x2
u(0,x)=ϕ(x),u(1,x)=u(2,x),x∈R,
is(seefordetails,Section2.2)
cos(ξt)[sinξ−sin(2ξ)]−sin(ξt)[cosξ−cos(2ξ)]
s(t,ξ)= ,
sinξ−sin(2ξ)
which has irreducible singularities at the points πk,k=±1,±2,..., and ±π/3+
2πm,m = 0,±1,.... Nonlocal boundary value problems arise, for instance, in
plasma physics, diffusionin porousmedia, etc. Pseudo-differentialoperatorswith
symbolssingularwith respectto dualvariables(ΨDOSS) allow us to revisitfrom
a new angle many classic bounda√ry value problems, as well. Examples include
the Schro¨dinger operator i∂ − ω I−ω2Δ, arising in the theory of relativisti-
∂t ch¯
cally free particles, the theory of Bessel potentials with symbols |ξ|−α, where α
is an arbitrary positive number. Many other examples, including fractional order
Preface ix
pseudo-differentialequationswithsingularsymbols,willbeconsideredthroughout
thebook.
Pseudo-differential operators with symbols singular with respect to dual
variables were first considered systematically by Cordes, Williams [CW77],
Plamenevski[Pla86],andJu.A.Dubinskii[Dub82],thoughepisodicallyappeared
earlier in worksof otherauthors(e.g.,[Kon67]).Cordesand Williamsused meth-
odsofabstractalgebratodefinealgebrasofpseudo-differentialoperatorswithnon-
smoothsymbolswith respecttodualvariablesξ=(ξ,...,ξ), andappliedthese
1 n
algebrastosolutionofsingularellipticpseudo-differentialequations.Plamenevski
used the Mellin’s transform approach to the construction of singular pseudo-
differential operators. Dubinskii defined his operators with the help of infinite
order differential operators. The latter requires the analyticity of the symbols in
domainsnotcontainingsingularities.Hence,thecorrespondingalgebrasofpseudo-
differential operators have locally analytic symbols. We also note that pseudo-
differentialoperatorswithsymbolssingularwithrespecttothecurrentvariablewere
studiedbyNagasein[Nag77].
Ourapproachisdifferentfromthosementionedabove.InChapter 2ofthisbook
weintroduceandstudyΨDOSSintheform
(cid:2)
1
Af(x)= a(x,ξ)F[f](ξ)e−i(x,ξ)dξ,
(2π)n
Rn
whereF[f](ξ)istheFouriertransformof f and(x,ξ)=x ξ +...,+x ξ.Dueto
1 1 n n
singularityofthesymbola(x,ξ),thisoperatorisnotwelldefinedevenoninfinitely
differentiable functions with compact support. Indeed, let f ∈C∞(Rn), such that
0 0
F[f ](ξ)=1inaneighborhoodofanon-integrablesingularpointξ ofthesymbol
0 0 0
a(x,ξ).ThentheintegralintheabovedefinitionoftheoperatorAdiverges.There-
fore,we introducea specialclass oftest functionsandthecorrespondingspaceof
distributions,forwhichtheoperatorAwithasingularsymbola(x,ξ)iswelldefined.
ThespaceoftestfunctionsdenotedbyΨG,p(Rn),whereG⊂Rnand p∈[1,∞),and
thecorrespondingspaceofdistributions(wecallthemψ-distributions)denotedby
Ψ(cid:5) (Rn),arerelativelynewandnotadequatelypresentedintheliterature.These
−G,p(cid:5)
spacesarestudiedindetailinChapter1.TheFouriertransformofanyfunctionin
ΨG,p(Rn), by definition, has a compact support contained in G. The dependence
of spacesΨG,p(Rn) andΨ−(cid:5)G,p(cid:5)(Rn) on the parameter p is not formal (see details
in Section 1.10). L. Ho¨rmander in his book [Hor83] pointed out that the Fourier
transform is not well adapted to L -spaces. However, in the above construction
p
onecannotavoidworkingwithFouriertransforminLp-spaces,sinceΨG,p(Rn)and
Ψ(cid:5) (Rn)dependinanessentialwayon p.
−G,p(cid:5)
Chapter 4 discusses the existence and uniqueness problem of general nonlo-
cal boundary value problems for ΨDOSS as well as their applications to vari-
ous problems of analysis. Initial and other types of boundary value problems for
fractional orderΨDOSS are important to include in our book due to their wide
applicationsin science and engineering.For instance, the fractionalorderFokker-
Planck-Kolmogorov(FPK) equationsare used as mathematicalmodels of various