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

Preface
Contents
Sobolev Spaces and Elliptic Boundary Value Problems
	1 Sobolev Spaces, Inequalities, Dirichlet, and Neumann Problems for the Laplacian
		1.1 Sobolev Spaces
		1.2 First Properties
		1.3 Traces
		1.4 Interpolation
		1.5 Transposition
		1.6 Inequalities
		1.7 Weak Solutions
		1.8 Strong Solutions
		1.9 Very Weak Solutions
		1.10 Solutions in Hs(Ω), with  0 < s < 2
	2 The Stokes Problem with Various Boundary Conditions
		2.1 The Problem (S) with Dirichlet Boundary Condition
		2.2 The Stokes Problem with Navier Type Boundary Condition
		2.3 The Stokes Problem with Navier Boundary Condition
	References
Survey on the Decay of the Local Energy for the Solutions of the Nonlinear Wave Equation
	1 Introduction and Preliminaries
	2 Scattering for the Subcritical and Critical Wave Equation
		2.1 The Subcritical Case
			2.1.1 Prisized Morawetz Estimate
			2.1.2 Global Time Strichartz Norms
			2.1.3 The Proof of Theorem 2.1
		2.2 The Critical Case
			2.2.1 Global Time Strichartz Norms
			2.2.2 The Proof of Theorem 2.1 in the Case p=5
	3 Exponential Decay for the Local Energy of the Subcritical and Critical Wave Equation with Localized Semilinearity
		3.1 Nonlinear Lax–Phillips Theory
		3.2 Exponential Decay for the Local Energy of the Subcritical Wave Equation
			3.2.1 The Compactness of Z(T)
			3.2.2 Proof of Theorem 3.1
		3.3 Exponential Decay for the Local Energy of the Critical Wave Equation
	4 Polynomial Decay for the Local Energy of the Semilinear Wave Equation with Small Data
		4.1 Fundamental Lemmas
		4.2 Proof of Theorem 4.1: Existence and Decay of the Local Energy
	5 Decay of the Local Energy for the Solutions of the Critical Klein–Gordon Equation
		5.1 Strichartz Norms Global in Time
		5.2 Exponential Decay of the Local Energy of Localized Linear Klein–Gordon Equation
			5.2.1 Semi-Group of Lax–Phillips Adapted to Localized Linear Klein–Gordon Equation
			5.2.2 Proof of Theorem 5.9
		5.3 Proof of Theorem 5.1
	Appendix
	References
A Spectral Numerical Method to Approximate the Boundary Controllability of the Wave Equation with Variable Coefficients
	1 Introduction
	2 Numerical Approximation of the Control Problem
	3 Minimal L2-Weighted Controls
	4 Numerical Experiments
	5 Appendix
	References
Aggregation Equation and Collapse to Singular Measure
	1 Introduction
	2 Graph Reformulation and Main Results
	3 Dini and Hölder Spaces
	4 Modified Curved Cauchy Operators
	5 Local Well-Posedness
	6 Global Well-Posedness
		6.1 Weak and Strong Damping Behavior of the Source Term
		6.2 Global a Priori Estimates
	References
Geometric Control of Eigenfunctions of Schrödinger Operators
	1 Introduction
	2 The Geometric Control Condition
	3 Are There Examples for Which (OE(ω)) Holds and (OS(ω)) Does Not?
	4 A Geometric Interpretation of (V-GCC) and Proof of Theorem 9
	5 On the Proof of Theorem 10
	References
Stability of a Graph of Strings with Local Kelvin–Voigt Damping
	1 Introduction
	2 Well-Posedness of the System
	3 Asymptotic Behavior
	References