Distribution Theory Applied to Differential Equations

277 Pages·2021·3.029 MB·English

by  Adina Chirila, Marin Marin, Andreas Öchsner

Adina Chirilă
Marin Marin
Andreas Öchsner
Distribution
Theory Applied
to Differential
Equations
Distribution Theory Applied to Differential
Equations
ă Ö
Adina Chiril Marin Marin Andreas chsner
(cid:129) (cid:129)
Distribution Theory Applied
to Differential Equations
123
AdinaChirilă Marin Marin
Department ofMathematics Department ofMathematics
andComputer Sciences andComputer Sciences
Transilvania University of Braşov Transilvania University of Braşov
Braşov,Romania Braşov,Romania
Andreas Öchsner
Faculty of MechanicalEngineering
Esslingen University of Applied Sciences
Esslingen am Neckar,Baden-Württemberg
Germany
ISBN978-3-030-67158-7 ISBN978-3-030-67159-4 (eBook)
https://doi.org/10.1007/978-3-030-67159-4
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Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Initial Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Test Functions and Regularization . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Seminorms and Locally Convex Spaces. . . . . . . . . . . . . . . . . . 11
2.3.1 Locally Convex Spaces. . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 Convex and Balanced Sets . . . . . . . . . . . . . . . . . . . . . 13
2.3.3 Absorbing Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Duals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 Reflexive Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 The Inductive Limit Topology. . . . . . . . . . . . . . . . . . . . . . . . . 20
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Convex and Lower-Semicontinuous Functions . . . . . . . . . . . . . . . . 25
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Convex Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Lower Semicontinuous Functions. . . . . . . . . . . . . . . . . . . . . . . 28
3.4 Convexity and Lower Semicontinuity. . . . . . . . . . . . . . . . . . . . 30
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 The Subdifferential of a Convex Function. . . . . . . . . . . . . . . . . . . . 37
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 The Conjugate Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 The Additivity of the Subdifferential . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
v
vi Contents
5 Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 The Resolvent and the Yosida Approximation . . . . . . . . . . . . . 61
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6 Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.1 Fundamental Spaces in the Theory of Distributions. . . . . . . . . . 73
6.1.1 On Some Properties of the Spaces CmðXÞ
and C1ðXÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.2 The Space of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3 The Dual of C1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.4 The Derivative of a Distribution . . . . . . . . . . . . . . . . . . . . . . . 87
6.5 Distributions as Generalized Functions. . . . . . . . . . . . . . . . . . . 90
6.6 On Some Spaces of Distributions. . . . . . . . . . . . . . . . . . . . . . . 96
6.7 The Primitive of a Distribution . . . . . . . . . . . . . . . . . . . . . . . . 96
6.7.1 Structure Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.8 Extras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.8.1 Higher-Order Primitives . . . . . . . . . . . . . . . . . . . . . . . 109
6.8.2 The Local Structure of Distributions . . . . . . . . . . . . . . 112
6.9 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.9.1 The Direct Product of Distributions. . . . . . . . . . . . . . . 117
6.9.2 Convolution of Distributions. . . . . . . . . . . . . . . . . . . . 121
6.9.3 Convolution of Functions and Distributions:
Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.9.4 Convolution Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 Tempered Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.1 The Schwartz Space of Infinitely Differentiable Functions
Rapidly Decreasing at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2 Tempered Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.3 The Fourier Transform in SðRnÞ . . . . . . . . . . . . . . . . . . . . . . . 140
7.3.1 The Inverse Fourier Transform . . . . . . . . . . . . . . . . . . 142
7.3.2 Properties of the Fourier Transform. . . . . . . . . . . . . . . 144
7.4 Fourier Transform of Tempered Distributions. . . . . . . . . . . . . . 145
7.5 The Fourier Transform of a Distribution with Compact
Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.6 The Product of a Distribution by a C1 Function . . . . . . . . . . . 151
7.7 The Space of Multipliers of S0ðRnÞ . . . . . . . . . . . . . . . . . . . . . 152
7.8 Some Results on Convolutions with Tempered Distributions. . . 156
7.9 The Paley-Wiener-Schwartz Theorem. . . . . . . . . . . . . . . . . . . . 160
7.10 A Result on the Fourier Transform of a Convolution
of Two Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Contents vii
8 Differential Equations in Distributions . . . . . . . . . . . . . . . . . . . . . . 167
8.1 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 167
8.1.1 Linear Differential Equations with Constant
Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.1.2 An Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
8.2 Partial Differential Equations. . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.2.1 The Direct Product. . . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.2.2 Hyperbolic Partial Differential Equations . . . . . . . . . . . 183
8.2.3 Parabolic Partial Differential Equations . . . . . . . . . . . . 185
8.2.4 Elliptic Partial Differential Equations. . . . . . . . . . . . . . 186
8.2.5 The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.2.6 An Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.1 The Sobolev Space H1ðXÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.2 The Sobolev Space HmðXÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.3 The Sobolev Space Wk;pðXÞ . . . . . . . . . . . . . . . . . . . . . . . . . . 203
9.4 The Sobolev Spaces HsðRnÞ . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.5 Besov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
9.5.1 The Nonhomogeneous Littlewood-Paley
Decomposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
9.5.2 Definition and Properties . . . . . . . . . . . . . . . . . . . . . . 217
9.5.3 The Homogeneous Littlewood-Paley Decomposition
and the Homogeneous Besov Spaces. . . . . . . . . . . . . . 222
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
10 Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
10.1.1 The Stokes System. . . . . . . . . . . . . . . . . . . . . . . . . . . 230
10.1.2 The Elasticity System. . . . . . . . . . . . . . . . . . . . . . . . . 232
10.1.3 The Plate Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . 235
10.2 The Approximation of Variational Problems. . . . . . . . . . . . . . . 237
10.2.1 The Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . 241
10.2.2 The Finite Element Method . . . . . . . . . . . . . . . . . . . . 242
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
11 On Some Spaces of Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 249
11.1 The Spaces DLp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
11.2 The Space O0C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
viii Contents
12 On Some Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
12.1 Local and Pseudolocal Operators . . . . . . . . . . . . . . . . . . . . . . . 263
12.2 Hypoelliptic Partial Differential Operators . . . . . . . . . . . . . . . . 266
12.3 Existence of Fundamental Solutions. . . . . . . . . . . . . . . . . . . . . 267
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 275
Chapter 1
Introduction
Abstract We give a brief motivation for the concept of distributions. We explain
why this concept is useful in applied mathematics. We present some landmarks in
thehistoryofthissubject.Thenwepresentthemaintopicsandresultsinthetheory
ofdistributions.
1.1 InitialRemarks
Theconceptofdistributionsisarelativelyrecentonesinceitwasusedforthefirst
timein1951inastudyoftheFrenchmathematicianL.Schwarz[43].Distributions
are a generalization of the concept of functions, hence some mathematicians call
themtoday“generalizedfunctions”.Thisgeneralizationwasmotivatedbypractical
situations. For instance, a boundary value problem from the theory of differential
equations (or even a Cauchy problem) has a solution if the right-hand side of the
problemisatleastofclassC1.Butinthephenomenamodelledbytheseproblems
theseconditionsarenotsatisfied.Inmostofthesephenomena,thefunctionsarenot
evencontinuous,veryoftenhavingdiscontinuitiesofthesecondkind.
We will see that the same problem, formulated in the context of the theory of
distributions will not impose restrictions on the right-hand side, these being auto-
maticallysatisfied.Weanticipatethateverydistributionisdifferentiable(inasense
thatwewilldiscussandwhichdoesnotdiffertoomuchfromthedifferentiationof
afunction)ofeveryorder.Inthecontextofthetheoryofdistributionstheregularity
conditionsareweaker.Forexample,thefunctionsdonotneedtobecontinuous[1,
2, 4].
Thetheoryofdistributions,asitisstructuredtoday,hasthedisadvantagethatit
employsverysophisticatedmathematicalconcepts,seeforexample[13,18,20,21,
31,35,42].Therefore,thistheoryislessaccessibletotheresearchersinmathematics.
Mostoftheconceptsusedinthetheoryofdistributionsareattheintersectionofother
subjectsofmathematics.
Thefundamentalsofthetheoryofdistributionsarebasedonadvancedtopicsin
functionalanalysis[3,5,6,9,24,30,36,37,47,49],topology,mathematicalphysics
[7,10,11,14,16,17,19,22,23,25,32,34,38–40,45,46,48]differentialgeometry
©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2021 1
A.Chirila˘etal.,DistributionTheoryAppliedtoDifferentialEquations,
https://doi.org/10.1007/978-3-030-67159-4_1
2 1 Introduction
and so on [8, 15, 26–29, 33, 44]. We may consider that the origins of this theory
coincide with Sobolev’s trial to define generalized solutions for hyperbolic partial
differentialequationsinthe1940s.Sobolev’sideawasemployedbyL.Schwarzin
the 1960s, when the concept of distributions was used for the first time. In 1971
Schwarz[43]publishedabookinwhichhelaidthefoundationsoftheunifiedtheory
ofdistributionsbyusingthemethodoflinearfunctionals.Afterwards,Sikorski[41]
introducedatheoryofdistributionsbyusingfundamentalsequencesofcontinuous
functions.ThismethodissimilartothatusedbyDedekind[12]tointroducethereal
numbersascutsinthesetofnaturalnumbers.
The wayinwhichSchwarz introduced theconcept of distributionsseemsmore
naturalandforthisreasonweuseitinthesequel.SincethecontributionsofSchwarz
intersect with the ones of Sobolev, many concepts and results from the theory of
distributions use the names of these two mathematicians, even though the theory
as it is today used, is almost completely different from that initiated by these two
scientists.
Themainqualitativeimprovementexpectedfromthistheoryisthesimplification
of the mathematical theory in order to make it more accesible. Starting with the
initiatorsofthistheory,theconceptshavebecomemorecomplex.
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