Preface
Contents
Contributors
On Friedrichs Commutators Lemma for Hardy Spacesand Applications
1 Introduction
2 Some Background on Local Hardy Spaces
3 Multipliers in hp(RN)
4 Friedrichs Lemma in hp(RN)
5 Elliptic Regularity in Hardy-Sobolev Spaces
References
On the Hardy Constant of Some Non-convex Planar Domains
1 Introduction
2 Auxiliary Results
3 Proof of Theorem 1.1
4 Domains Eβ,γ with Two Non-convex Angles
5 A Dirichlet–Neumann Hardy Inequality
References
Sharp Singular Trudinger-Moser-Adams Type Inequalities with Exact Growth
1 Introduction
2 Some Useful Results
2.1 Rearrangements
3 Sharp Singular Truding-Moser Type Inequality with Exact Growth
4 Sharp Singular Adams Type Inequality with Exact Growth
5 Proof of Theorem 1.4
6 Sharpness of Constants in Theorems 1.1 and 1.2
References
A Quantitative Lusin Theorem for Functions in BV
1 Introduction
2 Proof: A Partition of the Cube
3 Proof of the Lemma 1.1 When N=2
4 Proof of the Lemma 1.1 When N>2
References
X-Elliptic Harmonic Maps
1 Statement of Main Results
2 X-Elliptic Operators: Structural Assumptions
3 First Variation Formula
4 X-Elliptic Harmonic Maps into Spheres
5 X-Elliptic Harmonic Morphisms
References
Sum Operators and Fefferman–Phong Inequalities
1 Introduction
2 Preliminaries
3 Embedding Inequality of Fefferman-Phong Type
4 Unique Continuation
References
Lp-Parabolic Regularity and Non-degenerate Ornstein-Uhlenbeck Type Operators
1 Introduction and Basic Notations
2 A-priori Lp-Estimates
2.1 Proof of Theorem 2.1 When p=2
2.2 Proof of Theorem 2.1 When 1 < p < ∞
2.2.1 The Stochastic Integral
2.2.2 Proof of the Theorem
3 Lp-Estimates Involving Ornstein-Uhlenbeck Operators
References
Local Solvability of Nonsmooth Hörmander's Operators
1 Introduction and Main Results
2 The Classical Framework
2.1 Homogeneous Groups
2.2 Lifting and Approximation
3 Attacking the Problem: The Nonsmooth Background
4 The Parametrix Method
5 Local Solvability
6 Real Analysis Estimates
References
Multiple Solutions for an Eigenvalue Problem Involving Non-local Elliptic p-Laplacian Operators
1 Introduction
2 Preliminaries and Auxiliary Results
3 The Main Result
References
Uniqueness of Solutions of a Class of Quasilinear SubellipticEquations
1 Introduction
2 Notations and Definitions
3 A Priori Estimates
4 Comparison and Uniqueness
5 Further Applications
5.1 Symmetry Results
5.2 Some Applications to Systems
5.3 An Interesting Question
6 Inequalities and M-p-C Operators
7 Examples
References
Liouville Type Theorems for Non-linear Differential Inequalities on Carnot Groups
1 Introduction
2 A Brief History of the Keller-Osserman Condition
3 Coercive Differential Inequalities on Carnot Groups
4 A Sharper Result for the Heisenberg Group
References
Modica Type Gradient Estimates for Reaction-Diffusion Equations
1 Introduction
2 Forward Modica Type Estimates in Rn [0, T] for the Generalized Motion by Mean Curvature Equation
3 Forward Gradient Bounds for the Reaction-Diffusion Equation (5) in Epigraphs
4 Gradient Estimates for the Reaction-Diffusion Equation (5) in Rn (-∞,0] and Ω(-∞,0]
5 Modica Type Estimates for Reaction-Diffusion Equations on Compact Manifolds with Nonnegative Ricci Tensor
6 On a Conjecture of De Giorgi and Level Sets of Solutions to (5)
7 A Parabolic Version of the Conjecture of De Giorgi
References
A Few Recent Results on Fully Nonlinear PDE's
1 Introduction
2 Glaeser's Type Inequalities
2.1 An n-Dimensional Non-Smooth Glaeser Type Inequality
2.2 Non-negative Functions in Bounded Sets
3 Entire Subsolutions
References
Hölder Regularity of the Gradient for Solutions of Fully Nonlinear Equations with Sub Linear First Order Term
1 Introduction
2 Interior Regularity Results
2.1 Holder Regularity of the Gradient: Main Ingredients
2.2 Holder Regularity of the Gradient: Conclusion
Appendix
References
The Reflector Problem and the Inverse Square Law
1 Introduction
2 Ellipsoids
3 Reflectors and Reflector Measures
3.1 Physical Visibility Issues
4 Solution of the Problem in the Discrete Case
4.1 Discussion About Overshooting in the Discrete Case
5 Solution for a General Measure μ
5.1 Discussion About Overshooting
6 The Differential Equation for the Problem
References
Gagliardo-Nirenberg Inequalities for Horizontal Vector Fields in the Engel Group and in the Seven-Dimensional Quaternionic Heisenberg Group
1 Introduction
2 Preliminary Results and Differential Forms in Carnot Groups
3 Engel Group
4 The Seven-Dimensional Quaternionic Heisenberg Group
References
Regularity of the Free Boundary in Problems with DistributedSources
1 Main Definitions and Results
2 Reduction of Theorem 1.2 to a Localized Form
3 Lipschitz Implies C1,γ
4 The Nondegenerate Case
4.1 Improvement of Flatness
4.2 The Linearized Problem
4.3 Compactness
5 The Degenerate Case
5.1 Improvement of Flatness
6 Remarks and Further Developments
References
The Role of Fundamental Solution in Potential and Regularity Theory for Subelliptic PDE
1 Introduction
1.1 Applications to Complex Analysis, Finance and Vision
2 Fundamental Solutions of Linear Operators
2.1 Kolmogorov Type Operators
2.2 Gaussian Estimates for the Fundamental Solution of Heat Operators
2.3 Fundamental Solution of More General Operators
3 Balls, Mean Value Formulas and Potential Theory
3.1 Almost Exponential Maps and Poincaré Inequality
3.2 Mean Value Formulas on Level Sets and Poincaré Inequality
3.3 L-Subharmonicity and Average Operators
4 Global Regularity Results
4.1 A First Regularity Result at the Boundary
4.2 A Global Regularity Result in Lp Spaces
5 Non Linear Curvature Equations
5.1 Regularity Results for the Levi Equations
5.2 A Negative Regularity Result
References