Partial Differential Equations (Cornerstones)

Contents
Preface
	Preface to the Third Edition
	Preface to the Second Edition
	Preface to the First Edition
0 PRELIMINARIES
	1 Green’s Theorem
		1.1 Differential Operators and Adjoints
	2 The Continuity Equation
	3 The Heat Equation and the Laplace Equation
		3.1 Variable Coefficients
	4 A Model for the Vibrating String
	5 Small Vibrations of a Membrane
	6 Transmission of Sound Waves
	7 The Navier–Stokes System
	8 The Euler Equations
	9 Isentropic Potential Flows
		9.1 Steady Potential Isentropic Flows
	10 Partial Differential Equations
	Problems and Complements
		3c The Heat Equation and the Laplace Equation
			3.1c Basic Physical Assumptions
			3.2c The Diffusion Equation
			3.3c Justifying the Postulates (3.3c)–(3.4c)
			3.4c More on the Postulates (3.3c)–(3.4c)
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
	1 Quasi-Linear Second-Order Equations in Two Variables
	2 Characteristics and Singularities
		2.1 Coefficients Independent of ux and uy
	3 Quasi-Linear Second-Order Equations
		3.1 Constant Coefficients
		3.2 Variable Coefficients
	4 Quasi-Linear Equations of Order m ≥ 1
		4.1 Characteristic Surfaces
	5 Analytic Data and the Cauchy–Kowalewski Theorem
		5.1 Reduction to Normal Form ([32])
	6 Proof of the Cauchy–Kowalewski Theorem
		6.1 Estimating the Derivatives of u at the Origin
	7 Auxiliary Inequalities
	8 Auxiliary Estimations at the Origin
	9 Proof of the Cauchy–Kowalewski Theorem (Concluded)
		9.1 Proof of Lemma 6.1
	10 Holmgren’s Uniqueness Theorem
	11 Proof of the Holmgren Uniqueness Theorem
		11.1 Proof of Lemma 11.1
	Problems and Complements
		1c Quasi-Linear Second-Order Equations in Two Variables
		5c Analytic Data and the Cauchy–Kowalewski Theorem
		6c Proof of the Cauchy–Kowalewski Theorem
		8c The Generalized Leibniz Rule
		9c Proof of the Cauchy–Kowalewski Theorem Concluded
2 THE LAPLACE EQUATION
	1 Preliminaries
		1.1 The Dirichlet and Neumann Problems
		1.2 The Cauchy Problem
		1.3 Well-Posedness and a Counterexample of Hadamard
		1.4 Radial Solutions
	2 The Green and Stokes Identities
		2.1 The Stokes Identities
	3 Green’s Function and the Dirichlet Problem for a Ball
		3.1 Green’s Function for a Ball
	4 Sub-Harmonic Functions and the Mean Value Property
		4.1 The Maximum Principle
		4.2 Structure of Sub-Harmonic Functions
	5 Estimating Harmonic Functions and Their Derivatives
		5.1 The Harnack Inequality and the Liouville Theorem
		5.2 Analyticity of Harmonic Functions
	6 The Dirichlet Problem
	7 About the Exterior Sphere Condition
		7.1 The Case N = 2 and ∂E Piecewise Smooth
		7.2 A Counterexample of Lebesgue for N = 3 ([163])
	8 The Poisson Integral for the Half Space
	9 Schauder Estimates of Newtonian Potentials
	10 Potential Estimates in Lp(E)
	11 Local Solutions
		11.1 Local Weak Solutions
	12 Inhomogeneous Problems
		12.1 On the Notion of Green’s Function
		12.2 Inhomogeneous Problems
		12.3 The Case f ∈ C∞ o (E)
		12.4 The Case f ∈ Cη (E)
	Problems and Complements
		1c Preliminaries
			1.1c Newtonian Potentials on Ellipsoids
			1.2c Invariance Properties
		2c The Green and Stokes Identities
		3c Green’s Function and the Dirichlet Problem for the Ball
			3.1c Separation of Variables
		4c Sub-Harmonic Functions and the Mean Value Property
			4.1c Reflection and Harmonic Extension
			4.2c The Weak Maximum Principle
			4.3c Sub-Harmonic Functions
				4.3.1c A More General Notion of Sub-Harmonic Functions
		5c Estimating Harmonic Functions
			5.1c Harnack-Type Estimates
			5.2c Ill Posed Problems. An Example of Hadamard
			5.3c Removable Singularities
		7c About the Exterior Sphere Condition
		8c Problems in Unbounded Domains
			8.1c The Dirichlet Problem Exterior to a Ball
		9c Schauder Estimates up to the Boundary ([222, 223])
		10c Potential Estimates in Lp(E)
			10.1c Integrability of Riesz Potentials
			10.2c Second Derivatives of Potentials
3 BOUNDARY VALUE PROBLEMS BY DOUBLE LAYER POTENTIALS
	1 The Double-Layer Potential
	2 On the Integral Defining the Double-Layer Potential
	3 The Jump Condition of W(∂E, xo; v) Across ∂E
	4 More on the Jump Condition Across ∂E
	5 The Dirichlet Problem by Integral Equations ([192])
	6 The Neumann Problem by Integral Equations ([192])
	7 The Green’s Function for the Neumann Problem
		7.1 Finding g(·; ·)
	8 Eigenvalue Problems for the Laplacean
		8.1 Compact Kernels Generated by Green’s Function
	9 Compactness of AF in Lp(E) for 1 ≤  p ≤ ∞
	10 Compactness of AΦ in Lp(E) for 1 ≤  p ≤ ∞
	11 Compactness of AΦ in L∞(E)
	Problems and Complements
		2c On the Integral Defining the Double-Layer Potential
		5c The Dirichlet Problem by Integral Equations
		6c The Neumann Problem by Integral Equations
		7c The Green’s Function for the Neumann Problem
			7.1c Constructing g(·; ·) for a Ball in R2 and R3
				7.1.1c The Case N = 2
				7.1.2c The Case N = 3
		8c Eigenvalue Problems
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
	1 Kernels in L2(E)
		1.1 Examples of Kernels in L2(E)
			1.1.1 Kernels in L2(∂E)
	2 Integral Equations in L2(E)
		2.1 Existence of Solutions for Small |λ|
	3 Separable Kernels
		3.1 Solving the Homogeneous Equations
		3.2 Solving the Inhomogeneous Equation
	4 Small Perturbations of Separable Kernels
		4.1 Existence and Uniqueness of Solutions
	5 Almost Separable Kernels and Compactness
		5.1 Solving Integral Equations for Almost Separable Kernels
		5.2 Potential Kernels Are Almost Separable
	6 Applications to the Neumann Problem
	7 The Eigenvalue Problem
	8 Finding a First Eigenvalue and Its Eigenfunctions
	9 The Sequence of Eigenvalues
		9.1 An Alternative Construction Procedure of the Sequence of Eigenvalues
	10 Questions of Completeness and the Hilbert–Schmidt Theorem
		10.1 The Case of K(x;·) ∈ L2(E) Uniformly in x
	11 The Eigenvalue Problem for the Laplacean
		11.1 An Expansion of the Green’s Function
	Problems and Complements
		2c Integral Equations
			2.1c Integral Equations of the First Kind
			2.2c Abel Equations ([2, 3])
			2.3c Solving Abel Integral Equations
			2.4c The Cycloid ([3])
			2.5c Volterra Integral Equations ([266, 267])
		3c Separable Kernels
			3.1c Hammerstein Integral Equations ([114])
		6c Applications to the Neumann Problem
		9c The Sequence of Eigenvalues
		10c Questions of Completeness
			10.1c Periodic Functions in RN
			10.2c The Poisson Equation with Periodic Boundary Conditions
		11c The Eigenvalue Problem for the Laplacean
5 THE HEAT EQUATION
	1 Preliminaries
		1.1 The Dirichlet Problem
		1.2 The Neumann Problem
		1.3 The Characteristic Cauchy Problem
	2 The Cauchy Problem by Similarity Solutions
		2.1 The Backward Cauchy Problem
	3 The Maximum Principle and Uniqueness (Bounded Domains)
		3.1 A Priori Estimates
		3.2 Ill Posed Problems
		3.3 Uniqueness (Bounded Domains)
	4 The Maximum Principle in RN
		4.1 A Priori Estimates
		4.2 About the Growth Conditions (4.3) and (4.4)
	5 Uniqueness of Solutions to the Cauchy Problem
		5.1 A Counterexample of Tychonov ([263])
	6 Initial Data in L1 loc(RN)
		6.1 Initial Data in the Sense of L1loc(RN)
	7 Remarks on the Cauchy Problem
		7.1 About Regularity
		7.2 Instability of the Backward Problem
	8 Estimates Near t = 0
	9 The Inhomogeneous Cauchy Problem
	10 Problems in Bounded Domains
		10.1 The Strong Solution
		10.2 The Weak Solution and Energy Inequalities
	11 Energy and Logarithmic Convexity
		11.1 Uniqueness for Some Ill Posed Problems
	12 Local Solutions
		12.1 Variable Cylinders
		12.2 The Case |α| = 0
	13 The Harnack Inequality
		13.1 Compactly Supported Sub-Solutions
		13.2 Proof of Theorem 13.1
			13.2.1 Locating the Supremum of u in Q1
			13.2.2 Positivity of u over a Ball
			13.2.3 Expansion of the Positivity Set
	14 Positive Solutions in ST
		14.1 Non-Negative Solutions
	Problems and Complements
		2c Similarity Methods
			2.1c The Heat Kernel Has Unit Mass
			2.2c The Porous Medium Equation
			2.3c The p-Laplacean Equation
			2.4c The Error Function
			2.5c The Appell Transformation ([10])
			2.6c The Heat Kernel by Fourier Transform
			2.7c Rapidly Decreasing Functions
			2.8c The Fourier Transform of the Heat Kernel
			2.9c The Inversion Formula
		3c The Maximum Principle in Bounded Domains
			3.1c The Blow-Up Phenomenon for Super-Linear Equations
				3.1.1c An Example for α = 2
			3.2c The Maximum Principle for General Parabolic Equations
		4c The Maximum Principle in RN
			4.1c Counterexamples of the Tychonov Type
		7c Remarks on the Cauchy Problem
		12c On the Local Behavior of Solutions
6 THE WAVE EQUATION
	1 The One-Dimensional Wave Equation
		1.1 A Property of Solutions
	2 The Cauchy Problem
	3 Inhomogeneous Problems
	4 A Boundary Value Problem (Vibrating String)
		4.1 Separation of Variables
		4.2 Odd Reflection
		4.3 Energy and Uniqueness
		4.4 Inhomogeneous Problems
	5 The Initial Value Problem in N Dimensions
		5.1 Spherical Means
		5.2 The Darboux Formula
		5.3 An Equivalent Formulation of the Cauchy Problem
	6 The Cauchy Problem in R3
	7 The Cauchy Problem in R2
	8 The Inhomogeneous Cauchy Problem
	9 The Cauchy Problem for Inhomogeneous Surfaces
		9.1 Reduction to Homogeneous Data on t = Φ
		9.2 The Problem with Homogeneous Data
	10 Solutions in Half Space. The Reflection Technique
		10.1 An Auxiliary Problem
		10.2 Homogeneous Data on the Hyperplane x3 = 0
	11 A Boundary Value Problem
	12 Hyperbolic Equations in Two Variables
	13 The Characteristic Goursat Problem
		13.1 Proof of Theorem 13.1: Existence
		13.2 Proof of Theorem 13.1: Uniqueness
		13.3 Goursat Problems in Rectangles
	14 The Noncharacteristic Cauchy Problem and the Riemann Function
	15 Symmetry of the Riemann Function
	Problems and Complements
		2c The d’Alembert Formula
		3c Inhomogeneous Problems
			3.1c The Duhamel Principle ([61])
		4c Solutions for the Vibrating String
		6c Cauchy Problems in R3
			6.1c Asymptotic Behavior
			6.2c Radial Solutions
			6.3c Solving the Cauchy Problem by Fourier Transform
				6.3.1c The 1-Dimensional Case
				6.3.2c The Case N = 3
		7c Cauchy Problems in R2 and the Method of Descent
			7.1c The Cauchy Problem for N = 4, 5
		8c Inhomogeneous Cauchy Problems
			8.1c The Wave Equation for the N and (N + 1)-Laplacean
				8.1.1c The Telegraph Equation
			8.2c Miscellaneous Problems
		10c The Reflection Technique
		11c Problems in Bounded Domains
			11.1c Uniqueness
			11.2c Separation of Variables
		12c Hyperbolic Equations in Two Variables
			12.1c The General Telegraph Equation
		14c Goursat Problems
			14.1c The Riemann Function and the Fundamental Solution of the Heat Equation
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
	1 Quasi-Linear Equations
	2 The Cauchy Problem
		2.1 The Case of Two Independent Variables
		2.2 The Case of N Independent Variables
	3 Solving the Cauchy Problem
		3.1 Constant Coefficients
		3.2 Solutions in Implicit Form
	4 Equations in Divergence Form and Weak Solutions
		4.1 Surfaces of Discontinuity
		4.2 The Shock Line
	5 The Initial Value Problem
		5.1 Conservation Laws
	6 Conservation Laws in One Space Dimension
		6.1 Weak Solutions and Shocks
		6.2 Lack of Uniqueness
	7 Hopf Solution of The Burgers Equation
	8 Weak Solutions to (6.4) When a(·) is Strictly Increasing
		8.1 Lax Variational Solution
	9 Constructing Variational Solutions I
		9.1 Proof of Lemma 9.1
	10 Constructing Variational Solutions II
	11 The Theorems of Existence and Stability
		11.1 Existence of Variational Solutions
		11.2 Stability of Variational Solutions
	12 Proof of Theorem 11.1
		12.1 The Representation Formula (11.4)
		12.2 Initial Datum in the Sense of L1 loc(R)
		12.3 Weak Forms of the PDE
	13 The Entropy Condition
		13.1 Entropy Solutions
		13.2 Variational Solutions of (6.4) Are Entropy Solutions Proposition
		13.3 Remarks on the Shock and the Entropy Conditions
	14 The Kruzhkov Uniqueness Theorem
		14.1 Proof of the Uniqueness Theorem I
		14.2 Proof of the Uniqueness Theorem II
		14.3 Stability in L1(RN)
	15 The Maximum Principle for Entropy Solutions
	Problems and Complements
		3c Solving the Cauchy Problem
		6c Explicit Solutions to the Burgers Equation
			6.2c Invariance of Burgers Equations by Some Transformation of Variables
			6.3c The Generalized Riemann Problem
		13c The Entropy Condition
		14c The Kruzhkov Uniqueness Theorem
8 NONLINEAR EQUATIONS OF FIRST ORDER
	1 Integral Surfaces and Monge’s Cones
		1.1 Constructing Monge’s Cones
		1.2 The Symmetric Equation of Monge’s Cones
	2 Characteristic Curves and Characteristic Strips
		2.1 Characteristic Strips
	3 The Cauchy Problem
		3.1 Identifying the Initial Data p(0, s)
		3.2 Constructing the Characteristic Strips
	4 Solving the Cauchy Problem
		4.1 Verifying (4.3)
		4.2 A Quasi-Linear Example in R2
	5 The Cauchy Problem for the Equation of Geometrical Optics
		5.1 Wave Fronts, Light Rays, Local Solutions and Caustics
	6 The Initial Value Problem for Hamilton–Jacobi Equations
	7 The Cauchy Problem in Terms of the Lagrangian
	8 The Hopf Variational Solution
		8.1 The First Hopf Variational Formula
		8.2 The Second Hopf Variational Formula
	9 Semigroup Property of Hopf Variational Solutions
	10 Regularity of Hopf Variational Solutions
	11 Hopf Variational Solutions (8.3) Are Weak Solutions of the Cauchy Problem (6.4)
	12 Some Examples
		12.1 Example I
		12.2 Example II
		12.3 Example III
	13 Uniqueness
	14 More on Uniqueness and Stability
		14.1 Stability in Lp(RN) for All p ≥ 1
		14.2 Comparison Principle
	15 Semi-Concave Solutions of the Cauchy Problem
		15.1 Uniqueness of Semi-Concave Solutions
	16 A Weak Notion of Semi-Concavity
	17 Semi-Concavity of Hopf Variational Solutions
		17.1 Weak Semi-Concavity of Hopf Variational Solutions Induced by the Initial Datum uo
		17.2 Strictly Convex Hamiltonian
	18 Uniqueness of Weakly Semi-Concave Variational Hopf Solutions
9 LINEAR ELLIPTIC EQUATIONS WITH MEASURABLE COEFFICIENTS
	1 Weak Formulations and Weak Derivatives
		1.1 Weak Derivatives
	2 Embeddings of W1,p(E)
		2.1 Compact Embeddings of W1,p(E)
	3 Multiplicative Embeddings of Wo1,p(E) and W 1,p(E)
		3.1 Some Consequences of the Multiplicative Embedding Inequalities
	4 The Homogeneous Dirichlet Problem
	5 Solving the Homogeneous Dirichlet Problem (4.1) by the Riesz Representation Theorem
	6 Solving the Homogeneous Dirichlet Problem (4.1) by Variational Methods
		6.1 The Case N = 2
		6.2 Gâteaux Derivative and The Euler Equation of J(·)
	7 Solving the Homogeneous Dirichlet Problem (4.1) by Galerkin Approximations
		7.1 On the Selection of an Orthonormal System in Wo1,2 (E)
		7.2 Conditions on f and f for the Solvability of the Dirichlet Problem (4.1)
	8 Traces on ∂E of Functions in W1,p(E)
		8.1 The Segment Property
		8.2 Defining Traces
		8.3 Characterizing the Traces on ∂E of Functions in W1,p(E)
	9 The Inhomogeneous Dirichlet Problem
	10 The Neumann Problem
		10.1 A Variant of (10.1)
	11 The Eigenvalue Problem
	12 Constructing The Eigenvalues of (11.1)
	13 The Sequence of Eigenvalues and Eigenfunctions
	14 A Priori L∞(E) Estimates for Solutions of the Dirichlet Problem (9.1)
	15 Proof of Propositions 14.1–14.2
		15.1 An Auxiliary Lemma on Fast Geometric Convergence
		15.2 Proof of Proposition 14.1 for N > 2
		15.3 Proof of Proposition 14.1 for N = 2
	16 A Priori L∞(E) Estimates for Solutions of the Neumann Problem (10.1)
	17 Proof of Propositions 16.1–16.2
		17.1 Proof of Proposition 16.1 for N > 2
		17.2 Proof of Proposition 16.1 for N = 2
	18 Miscellaneous Remarks on Further Regularity
	Problems and Complements
		1c Weak Formulations and Weak Derivatives
			1.1c The Chain Rule in W1,p(E)
		2c Embeddings of W1,p(E)
			2.1c Proof of (2.4)
			2.2c Compact Embeddings of W1,p(E)
		3c Multiplicative Embeddings of Wo1,p(E) and W1,p(E)
			3.1c Proof of Theorem 3.1 for 1 ≤ p < N
			3.2c Proof of Theorem 3.1 for p ≥ N > 1
				3.2.1c Estimate of I1(x,R)
				3.2.2c Estimate of I2(x,R)
				3.2.3c Proof of Theorem 3.1 for p ≥ N > 1 (Concluded)
			3.3c Proof of Theorem 3.2 for 1 ≤ p < N and E Convex
		5c Solving the Homogeneous Dirichlet Problem (4.1) by the Riesz Representation Theorem
		6c Solving the Homogeneous Dirichlet Problem (4.1) by Variational Methods
			6.1c More General Variational Problems
				A Prototype Example
				Lower Semi-Continuity
			6.8c Gâteaux Derivatives, Euler Equations and Quasi-Linear Elliptic Equations
				6.8.1c Quasi-Linear Elliptic Equations
				6.8.2c Quasi-Minima
		8c Traces on ∂E of Functions in W1,p(E)
			8.1c Extending Functions in W1,p(E)
			8.2c The Trace Inequality
			8.3c Characterizing the Traces on ∂E of Functions in W1,p(E)
		9c The Inhomogeneous Dirichlet Problem
			9.1c The Lebesgue Spike
			9.2c Variational Integrals and Quasi-Linear Equations
		10c The Neumann Problem
		11c The Eigenvalue Problem
		12c Constructing the Eigenvalues
		13c The Sequence of Eigenvalues and Eigenfunctions
		14c A Priori L∞(E) Estimates for Solutions of the Dirichlet Problem (9.1)
		15c A Priori L∞(E) Estimates for Solutions of the Neumann Problem (10.1)
			15.1c Back to the Quasi-Linear Dirichlet Problem (9.1c)
10 DEGIORGI CLASSES
	1 Quasi-Linear Equations and DeGiorgi Classes
		1.1 DeGiorgi Classes
	2 Local Boundedness of Functions in the DeGiorgi Classes
		2.1 Proof of Theorem 2.1 for 1 < p < N
		2.2 Proof of Theorem 2.1 for p = N
	3 Hölder Continuity of Functions in the DG Classes
		3.1 On the Proof of Theorem 3.1
	4 Estimating the Values of u by the Measure of the Set Where u Is Either Near μ+ or Near μ−
	5 Reducing the Measure of the Set Where u is Either Near μ+ or Near μ−
		5.1 The Discrete Isoperimetric Inequality
		5.2 Proof of Proposition 5.1
	6 Proof of Theorem 3.1
	7 Boundary DeGiorgi Classes: Dirichlet Data
		7.1 Continuity up to ∂E of Functions in the Boundary DG Classes (Dirichlet Data)
	8 Boundary DeGiorgi Classes: Neumann Data
		8.1 Continuity up to ∂E of Functions in the Boundary DG Classes (Neumann Data)
	9 The Harnack Inequality
		9.1 Proof of Theorem 9.1. Preliminaries
		9.2 Proof of Theorem 9.1. Expansion of Positivity Proposition 9.1
		9.3 Proof of Theorem 9.1
	10 Harnack Inequality and H¨older Continuity
	11 Local Clustering of the Positivity Set of Functions in W1,1(E)
	12 A Proof of the Harnack Inequality Independent of Hölder Continuity
11 LINEAR PARABOLIC EQUATIONS IN DIVERGENCE FORM WITH MEASURABLE COEFFICIENTS
	1 Parabolic Spaces and Embeddings
		1.1 Steklov Averages
	2 Weak Formulations
	3 The Homogeneous Dirichlet Problem
	4 The Energy Inequality
	5 Existence of Solutions of the Homogeneous Cauchy–Dirichlet Problem (3.1) by Galerkin Approximations
	6 Uniqueness of Solutions of the Homogeneous Cauchy–Dirichlet Problem (3.1)
	7 Traces of Functions on Σ def = ∂E × (0, T]
	8 The Inhomogeneous Dirichlet Problem
	9 The Neumann Problem
		9.1 The Energy Inequality for the Neumann Problem
		9.2 A Variant of Problems (3.1) and (9.1)
	10 A Priori L∞(ET ) Estimates for Solutions of the Cauchy–Dirichlet Problem (8.1)
	11 Proof of Propositions 10.1–10.2
	12 A Priori L∞(ET ) Estimates for Solutions of the Neumann Problem (9.1)
	13 Proof of Propositions 12.1–12.2
	14 Miscellaneous Remarks on Further Regularity
	15 Gaussian Bounds on the Fundamental Solution
		15.1 The Gaussian Upper Bound
		15.2 The Gaussian Lower Bound
	Problems and Complements
		3c The Homogeneous Dirichlet Problem
		5c Existence of Solutions of the Homogeneous Dirichlet Problem (3.1) by Galerkin Approximations
		7c Traces of Functions on Σdef= ∂E × (0, T]
		8c The Inhomogeneous Dirichlet Problem
			8.1c Parabolic Quasi-Minima
		9c The Neumann Problem
		10c A Priori L∞(ET ) Estimates for Solutions of the Dirichlet Problem (8.1)
		12c A Priori L∞(ET ) Estimates for Solutions of the Neumann Problem (9.1)
		15c Gaussian Bounds on the Fundamental Solution
12 PARABOLIC DEGIORGI CLASSES
	1 Quasi-Linear Equations and DeGiorgi Classes
		1.1 Parabolic DeGiorgi Classes
	2 Local Boundedness of Functions in the PDG Classes
	3 Hölder Continuity of Functions in the PDG Classes
		3.1 On the Proof of Theorem 3.1
	4 Estimating the Values of u by the Measure of the Set Where u is Either Near μ+ or Near μ−
	5 Reducing the Measure of the Set Where u is Either Near μ+ or Near μ−
		5.1 Proof of Proposition 5.1
	6 Propagating in Time the Measure-Theoretical Information
		6.1 Proof of Proposition 6.1
	7 Proof of Theorem 3.1
	8 Boundary Parabolic DeGiorgi Classes: Dirichlet Data
		8.1 Lateral Conditions
		8.2 Initial Conditions
		8.3 Definition of Boundary Parabolic DeGiorgi Classes
		8.4 Continuity up to ∂pET of Functions in the Boundary PDG Classes (Dirichlet Data)
	9 Boundary Parabolic DeGiorgi Classes: Neumann Data
		9.1 Lateral Boundary
		9.2 Definition of Boundary Parabolic DeGiorgi Classes
		9.3 Continuity up to ST of Functions in the Boundary PDG Classes (Neumann Data)
	10 The Harnack Inequality
		10.1 Proof of Theorem 10.1. Preliminaries
		10.2 Proof of Theorem 10.1. Expansion of Positivity
		10.3 Proof of Theorem 10.1
			10.3.1 Local Largeness of w Near (y, s)
			10.3.2 Expanding the Positivity of w
			10.3.3 Proof of Theorem 10.1 Concluded
		10.4 The Mean Value Harnack Inequality
			10.4.1 There Exists t < t o Satisfying (3.1)
	11 The Harnack Inequality Implies the Hölder Continuity
	12 A Consequence of the Harnack Inequality
	13 A More Straightforward Proof of the Hölder Continuity
	Problems and Complements
		2c Local Boundedness of Functions in the PDG Classes
		3c Hölder Continuity of Solutions of Linear Parabolic Equations with Bounded and Measurable Coefficients
		6c Propagating in Time the Measure-Theoretical Information
			6.1c Proof of Proposition 6.1c
		7c Proof of Theorem 3.1
		11c The Harnack Inequality
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
	1 Introductory Material
		1.1 Introduction
			1.1.1 Linear Equations
			1.1.2 Quasi-linear Equations
			1.1.3 Fully Nonlinear Equations
		1.2 The Pucci Equation
		1.3 The Bellman–Dirichlet Equation
		1.4 Remarks on the Concept of Ellipticity
		1.5 Equations of Mini-Max Type
	2 Maximum Principles
		2.1 Linear Equations
			2.1.1 The Dirichlet Problem
			2.1.2 The Neumann Problem
		2.2 Quasi-Linear Equations
			2.2.1 The Dirichlet Problem
			2.2.2 Variational Boundary Data
	3 The Aleksandrov Maximum Principle
		3.1 Basic Geometric Notions
			3.1.1 The Upper Contact Set
			3.1.2 The Concave Hull
			3.1.3 The Normal Mapping
			3.1.4 The Normal Mapping of a Cone
		3.2 Increasing Concave Hull of u
			3.2.1 Proof of Proposition 3.1
			3.2.2 Proof of Proposition 3.2
		3.3 Auxiliary Lemmas
		3.4 Embedding by Normal Mapping
		3.5 Estimates of the Supremum of a Function
		3.6 Maximum Principle for Nonlinear Operators
	4 Local Estimates and the Harnack Inequality
		4.1 A Local Maximum Principle
		4.2 A Covering Lemma
		4.3 Two Technical Lemmas
		4.4 The Harnack Inequality for Linear Equations
		4.5 The Harnack Inequality for Quasi-Linear Equations
		4.6 Local H¨older Continuity of Solutions
		4.7 Hölder Continuity of Solutions of Quasi-Linear Equations
	Problems and Complements
		1c Introductory Material
			1.1c Introduction
				1.1.1c Linear Equations
			1.3c The Bellman–Dirichlet Equation
		3c The Aleksandrov Maximum Principle
			3.5c Estimates of the Supremum of a Function
14 NAVIER–STOKES EQUATIONS
	1 Navier–Stokes Equations in Dimensionless Form
	2 Steady-State Flow with Homogeneous Boundary Data
		2.1 Uniqueness of Solutions to (2.1)
	3 Existence of Solutions to (2.1)
	4 Nonhomogeneous Boundary Data
		4.1 Uniqueness of Solutions to (4.1)
		4.2 Existence of Solutions to (4.1)
	5 Recovering the Pressure
	6 Steady-State Flows in Unbounded Domains
		6.1 Assumptions on a and f
		6.2 Toward a Notion of a Solution to (6.1)
	7 Existence of Solutions to (6.1)
		7.1 Approximating Solutions and A Priori Estimates
		7.2 The Limiting Process
	8 Time-Dependent Navier–Stokes Equations in Bounded Domains
	9 The Galerkin Approximations
	10 Selecting Subsequences Strongly Convergent in L2(ET; R3)
	11 The Limiting Process and Proof of Theorem 8.1
	12 Higher Integrability and Some Consequences
		12.1 The The Lp,q(ET ; RN) Spaces
		12.2 The Case N = 2
	13 Energy Identity for the Homogeneous Boundary Value Problem with Higher Integrability
	14 Stability and Uniqueness for the Homogeneous Boundary Value Problem with Higher Integrability
	15 Local Regularity of Solutions with Higher Integrability
	16 Proof of Theorem 15.1 – Introductory Results
	17 Proof of Theorem 15.1 Continued
	18 Proof of Theorem 15.1 Concluded
	19 Regularity of the Initial-Boundary Value Problem
	20 Recovering the Pressure in the Time-Dependent Equations
	Problems and Complements
		1c Navier–Stokes Equations in Dimensionless Form
		4c Nonhomogeneous Boundary Data
			4.1c Solving (4.1) by Galerkin Approximations
			4.2c Extending Fields a ∈ W 1/2 ,2(∂E; R3), Satisfying (4.2) into Solenoidal Fields b ∈ W1,2(E; R3)
			4.3c Proof of Proposition 4.3c
			4.4c The Case of a General Domain E
		5c Recovering the Pressure
			5.1c Proof of Proposition 5.1 for u ∈ H┴∩C∞(E; R3)
			5.2c Proof of Proposition 5.1 for u∈ H┴
			5.3c More General Versions of Proposition 5.1
		8c Time-Dependent Navier–Stokes Equations in Bounded Domains
		10c Selecting Subsequences Strongly Convergent in L2(ET )
			10.1c Proof of Friedrichs’ Lemma
			10.2c Compact Embedding of W1,p into Lq(Q) for 1 q < p*
			10.3c Solutions Global in Time
		11c The Limiting Process and Proof of Theorem 8.1
		12c Higher Integrability and Some Consequences
		13c Energy Identity for the Homogeneous Boundary Value Problem with Higher Integrability
		15c Local Regularity of Solutions with Higher Integrability
		16c Proof of Theorem 15.1 – Introductory Results
		20c Recovering the Pressure in the Time-Dependent Equations
15 QUASI-LINEAR FIRST-ORDER SYSTEMS
	1 Hyperbolic Systems
	2 Some Examples
		2.1 Incompressible Euler Equations
		2.2 Reacting Gas Flow in 1–Space Dimension
		2.3 A Weakly Hyperbolic System Arising in Magnetohydrodynamics
	3 Uniqueness of Smooth Solutions
	4 Existence of Solutions: The Linear Theory
		4.1 A Family of Approximating Problems
		4.2 Estimate of Hi, i = 1, 2, 3
		4.3 Proof of Theorem 4.1
	5 Existence of Solutions: The Nonlinear Theory
	6 An Interlude: Counterexamples to Uniqueness in the Linear Case
	7 Back to Quasi-Linear First-Order Strictly Hyperbolic Systems
		7.1 A First Example
		7.2 A Second Example
	8 Lax Shock Conditions
	9 Shocks
		9.1 An Example
	10 Centered Rarefaction Waves
		10.1 An Example
	11 Contact Discontinuities
		11.1 An Example
	12 The Riemann Problem
	13 Convex Entropies
		13.1 Examples of Entropies for 2 × 2 Systems
	14 The Glimm Existence Result
	15 Some Final Comments
	Problems and Complements
		2c Some Examples
		5c Existence of Solutions: The Nonlinear Theory
		6c Proof of Theorem 6.1
		7c Back to Quasi-Linear First-Order Strictly Hyperbolic Systems
		12c The Riemann Problem
		13c Convex Entropies
References
Index