M2edp

Université Aix Marseille
Master 2 de mathématiques
Equations aux dérivées partielles
Thierry Gallouët, Raphaèle Herbin
29 janvier 2018

Table des matières
1 Sobolev spaces 4
1.1 Weakderivatives ........................................... 4
1.2 Definitionandproperties ....................................... 6
1.3 FunctionalAnalysistools....................................... 8
1.4 Densitytheorems ........................................... 9
1.5 Théorèmesdetrace.......................................... 10
1.6 Théorèmesdecompacité ....................................... 11
1.7 Sobolevembeddings ......................................... 11
1.8 Exercices ............................................... 12
2 Problèmes elliptiques linéaires 30
2.1 Formulationfaible........................................... 30
2.2 Analysespectrale ........................................... 35
2.2.1 Quelquesrappels....................................... 35
2.2.2 LeLaplacien ......................................... 35
2.3 Régularitédessolutions........................................ 38
2.4 Positivité de la solution faible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Condition de Dirichlet non homogène . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.6 Exercices ............................................... 44
3 Problèmes elliptiques non linéaires 78
3.1 Méthodesdecompacité........................................ 78
3.1.1 Degré topologique et théorème de Schauder . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.1.2 Existence avec le théorème de Schauder . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.1.3 Existence avec le degré topologique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2 Méthodesdemonotonie........................................ 91
3.2.1 Introduction.......................................... 91
3.2.2 OpérateurdeLeray-Lions .................................. 92
3.3 Exercices ............................................... 101
4 Problèmes paraboliques 120
4.1 Aperçudesméthodes......................................... 120
4.2 Intégration à valeur vectorielle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.3 Existence Par Faedo-Galerkin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.4 problèmes paraboliques non linéaires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.5 Compacitéentemps.......................................... 154
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TABLE DES MATIÈRES TABLE DES MATIÈRES
4.6 Exercices ............................................... 167
4.7 Corrigésd’exercices ......................................... 173
5 Problèmes hyperboliques 188
5.1 Lecasunidimensionnel........................................ 188
5.2 Casmultidimensionnel ........................................ 199
5.3 Exercices ............................................... 206
5.4 Corrigésd’exercices ......................................... 210
Bibliography
EDP, Télé-enseignement, M2 2

Introduction
This course gives some tools for the study of partial differential equations. These tools are used to obtain existence
results (and also often uniqueness results) for some examples of linear or nonlinear PDEs of various nature (elliptic,
parabolic or hyperbolic).
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Chapitre 1
Sobolev spaces
Sobolev 1spaces are functional spaces, that is spaces containing functions, and these functions are such that their
powers and the powers of their derivatives (in the sense of transposition, or in a weak sense which we shall give
later) are Lebesgue-integrable. Similarly to the Lebesgue spaces, these spaces are Banach spaces 2. The fact that
the Sobolev spaces are complete is very important to prove the existence of solutions to some partial differential
equations.
1.1 Weak derivatives
The notion of weak derivative is already in a famous article by Jean Leray 3published in 1934, on the Navier-
Stokes equations ; in this article, it is named “quasi-derivative” ([7] page 205). This notion is crucial for the study
of the existence of solutions to many partial differential equations.
In the sequel, Ωis an open subset of IRN,N≥1, and C∞
c(Ω) denotes the set of functions of class C∞and with
compact support on Ω, that is :
C∞
c(Ω) = {u∈C∞(Ω); ∃K⊂Ω, K compact ;u= 0 sur Kc}.
Moreover, the open subset Ωwill always equipped with the Borel σ-algebra, denoted by B(Ω), and with the
Lebesgue measure, denoted by λif N= 1 and λNsi N > 1. Integration will always be with respect to the
Lebesgue measure, unless otherwise mentioned.
The following lemma is fundamental, because it allows to merge the (class of) function(s) f∈L1
loc(Ω) with the
linear mapping Tf:C∞
c(Ω) →IR defined by ϕ∈C∞ c(Ω) 7→ Tf(ϕ) = RΩf(x)ϕ(x) dx.
Recall that f∈L1
loc(Ω) means that for any compact subset Kof Ω, the restriction f|Kof fto Kbelongs to
L1(K).
Lemme 1.1 (Almost everywhere equality in L1)Let Ωbe an open subset of IRN,N≥1, and let fand g∈
L1
loc(Ω). Then :
∀ϕ∈C∞
c(Ω),ZΩ
f(x)ϕ(x) dx=ZΩ g(x)ϕ(x) dx⇐⇒ [f=ga.e..].
1. Sergueï Lvovitch Sobolev, 1908-1989) is a Russian mathematician and physicist.
2. A Banach space is a is a complete normed vector space.
3. Jean Leray, 1906 -1998, is a French mathematician who worked on partial differential equations and on algebraic topology.
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