An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs (Publications of the Scuola Normale Superiore)

Title Page
Copyright Page
Table of Contents
Preface to the first edition
Preface to the second edition
Chapter 1 Harmonic functions
	1.1 Introduction
	1.2 The variational method
		1.2.1 Non-existence of minimizers of variational integrals
		1.2.2 Non-finiteness of the Dirichlet integral
	1.3 Some properties of harmonic functions
	1.4 Existence in general bounded domains
		1.4.1 Solvability of the Dirichlet problem on balls: Poisson’s formula
		1.4.2 Perron’s method
		1.4.3 Poincar´e’s method
Chapter 2 Direct methods
	2.1 Lower semicontinuity in classes of Lipschitz functions
	2.2 Existence of minimizers
		2.2.1 Minimizers in Lipk(Ω)
		2.2.2 A priori gradient estimates
			Supersolutions and subsolutions
			The comparison principle
			Reduction to boundary estimates
			Boundary gradient estimates through the bounded slope condition
		2.2.3 Constructing barriers: the distance function
	2.3 Non-existence of minimizers
		2.3.1 An example of Bernstein
		2.3.2 Sharpness of the mean curvature condition
	2.4 Finiteness of the area of graphs with zero mean curvature
	2.5 The relaxed area functional in BV
		Functions of bounded variation
		2.5.1 BV minimizers for the area functional
Chapter 3 Hilbert space methods
	3.1 The Dirichlet principle
		The abstract Dirichlet’s principle
		The theorem of Riesz
		The projection theorem
		Bilinear symmetric forms
		The Lax-Milgram theorem
	3.2 Sobolev spaces
		3.2.1 Strong and weak derivatives
		3.2.2 Poincaré inequalities
		3.2.3 Rellich’s theorem
		3.2.4 The chain rule in Sobolev spaces
		3.2.5 The Sobolev embedding theorem
		3.2.6 The Sobolev-Poincaré inequality
	3.3 Elliptic equations: existence of weak solutions
		3.3.1 Dirichlet boundary condition
		3.3.2 Neumann boundary condition
	3.4 Elliptic systems: existence of weak solutions
		3.4.1 The Legendre and Legendre-Hadamard ellipticity conditions
		3.4.2 Boundary value problems for very strongly elliptic systems
		3.4.3 Strongly elliptic systems: Gårding’s inequality
Chapter 4 L2-regularity: the Caccioppoli inequality
	4.1 The simplest case: harmonic functions
	4.2 Caccioppoli’s inequality for elliptic systems
	4.3 The difference quotient method
		4.3.1 Interior L2-estimates
		4.3.2 Boundary regularity
	4.4 The hole-filling technique
Chapter 5 Schauder estimates
	5.1 The spaces of Morrey and Campanato
		5.1.1 A characterization of Hölder continuous functions
	5.2 Elliptic systems with constant coefficients: two basic estimates
		5.2.1 A generalization of Liouville’s theorem
	5.3 A lemma
	5.4 Schauder estimates for elliptic systems in divergence form
		5.4.1 Constant coefficients
		5.4.2 Continuous coefficients
		5.4.3 Hölder continuous coefficients
		5.4.4 Summary and generalizations
		5.4.5 Boundary regularity
	5.5 Schauder estimates for elliptic systems in non-divergence form
		5.5.1 Solving the Dirichlet problem
Chapter 6 Some real analysis
	6.1 The distribution function and an interpolation theorem
		6.1.1 The distribution function
		6.1.2 Riesz-Thorin’s theorem
		6.1.3 Marcinkiewicz’s interpolation theorem
	6.2 The maximal function and the Calderon-Zygmund argument
		6.2.1 The maximal function
			The maximal theorem of Hardy and Littlewood
			Lebesgue’s differentiation theorem
			A theorem of F. Riesz
		6.2.2 Calderon-Zygmund decomposition argument
	6.3 BMO
		6.3.1 John-Nirenberg lemma I
			Sobolev embedding in the limit case.
		6.3.2 John-Nirenberg lemma II
		6.3.3 Interpolation between Lp and BMO
		6.3.4 Sharp function and interpolation Lp − BMO
	6.4 The Hardy space H1
		6.4.1 The duality between H1 and BMO
	6.5 Reverse Hölder inequalities
		6.5.1 Gehring’s lemma
		6.5.2 Reverse Hölder inequalities with increasing support
Chapter 7 Lp-theory
	7.1 Lp-estimates
		7.1.1 Constant coefficients
		7.1.2 Variable coefficients: divergence and non-divergence case
		7.1.3 The cases p = 1 and p = ∞
		7.1.4 Wente’s result
	7.2 Singular integrals
		7.2.1 The cancellation property and the Cauchy principal value
		7.2.2 Hölder-Korn-Lichtenstein-Giraud theorem
		7.2.3 L2-theory
		7.2.4 Calderón-Zygmund theorem
	7.3 Fractional integrals and Sobolev inequalities
Chapter 8 The regularity problem in the scalar case
	8.1 Existence of minimizers by direct methods
	8.2 Regularity of critical points of variational integrals
		The associated quasilinear elliptic system
		The regularity of critical points of class C1.
		The C1-regularity of critical points
	8.3 De Giorgi’s theorem: essentially the original proof
		De Giorgi’s class
		The theorem and its proof
		A remark
	8.4 Moser’s technique and Harnack’s inequality
		The iteration technique
		The Harnack inequality
	8.5 Still another proof of De Giorgi’stheorem
	8.6 The weak Harnack inequality
	8.7 Differentiability of minimizers of non-differentiable variational integrals
Chapter 9 Partial regularity in the vector-valued case
	9.1 Counterexamples to everywhere regularity
		9.1.1 De Giorgi’s counterexample
		9.1.2 Giusti and Miranda’s counterexample
		9.1.3 The minimal cone of Lawson and Osserman
	9.2 Partial regularity
		9.2.1 Partial regularity of minimizers
		9.2.2 Partial regularity of solutions to quasilinear elliptic systems
		9.2.3 Partial regularity of solutions to quasilinear elliptic systems with quadratic right-hand side
		9.2.4 Partial regularity of minimizers of non-differentiable quadratic functionals
		9.2.5 The Hausdorff dimension of the singular set
Chapter 10 Harmonic maps
	10.1 Basic material
		10.1.1 The variational equations
			Outer variations
			Inner variations
		10.1.2 The monotonicity formula
	10.2 Giaquinta and Giusti’s regularity results
		10.2.1 The main regularity result
		10.2.2 The dimension reduction argument
	10.3 Schoen and Uhlenbeck’s regularity results
		10.3.1 The main regularity result
		10.3.2 The dimension reduction argument
			The compactness theorem
		10.3.3 The stratification of the singular set
			Tangent maps
	10.4 Regularity of 2-dimensional weakly harmonic maps
		10.4.1 Hélein’s proof when the target manifold is Sn
		10.4.2 Riviére’s proof for arbitrary target manifolds
			Proof of Theorem 10.41
			Proof of Theorem 10.48
		10.4.3 Irregularity of weakly harmonic maps in dimension 3 and higher
	10.5 Regularity of stationary harmonic maps
	10.6 The Hodge-Morrey decomposition
		10.6.1 Decomposition of differential forms
		10.6.2 Decomposition of vector fields
Chapter 11 A survey of minimal graphs
	11.1 Geometry of the submanifolds of Rn+m
		11.1.1 Riemannian structure and Levi-Civita connection
		11.1.2 The gradient, divergence and Laplacian operators
		11.1.3 Second fundamental form and mean curvature
		11.1.4 The area and its first variation
			First variation and mean curvature
			Definition of minimal surface
			The minimal surface system
		11.1.5 Area-decreasing maps
	11.2 Minimal graphs in codimension 1
		11.2.1 Convexity of the area; uniqueness and stability
			Uniqueness and minimizing properties
			Stability under parametric deformations
		11.2.2 The problem of Plateau: existence of minimal graphs with prescribed boundary
		11.2.3 A priori estimates
			Gradient estimates
		11.2.4 Regularity of Lipschitz continuous minimal graphs
		11.2.5 The a priori gradient estimate of Bombieri, De Giorgi and Miranda
		11.2.6 Regularity of BV minimizers of the area functional
	11.3 Regularity in arbitrary codimension
		11.3.1 Blow-ups, blow-downs and minimal cones
		11.3.2 Bernstein-type theorems
			Remarks on Bernstein’s theorem: the Gauss map
		11.3.3 Regularity of area-decreasing minimal graphs
		11.3.4 Regularity and Bernstein theorems for Lipschitz minimal graphs in dimension 2 and 3
	11.4 Geometry of Varifolds
		11.4.1 Rectifiable subsets of Rn+m
		11.4.2 Rectifiable varifolds
		11.4.3 First variation of a rectifiable varifold
		11.4.4 The monotonicity formula
		11.4.5 The regularity theorem of Allard
		11.4.6 Abstract varifolds
		11.4.7 Image and first variation of an abstract varifold
		11.4.8 Allard’s compactness theorem
Bibliography
Index
LECTURE NOTES
	Published volumes
	Volumes published earlier