Contemporary Research in Elliptic PDEs and Related Topics (Springer INdAM Series)

Preface
Contents
About the Editor
Getting Acquainted with the Fractional Laplacian
	1 The Laplace Operator
	2 Some Fractional Operators
		2.1 The Fractional Laplacian
		2.2 The Regional (or Censored) Fractional Laplacian
		2.3 The Spectral Fractional Laplacian
		2.4 Fractional Time Derivatives
	3 A More General Point of View: The ``Master Equation\'\'
	4 Probabilistic Motivations
		4.1 The Heat Equation and the Classical Laplacian
		4.2 The Fractional Laplacian and the Regional Fractional Laplacian
		4.3 The Spectral Fractional Laplacian
		4.4 Fractional Time Derivatives
		4.5 Fractional Time Diffusion Arising from Heterogeneous Media
	5 All Functions Are Locally s-Caloric (Up to a Small Error): Proof of (2.12)
	Appendix A: Confirmation of (2.7)
	Appendix B: Proof of (2.10)
	Appendix C: Proof of (2.14)
	Appendix D: Proof of (2.17)
	Appendix E: Deducing (2.19) from (2.15) Using a Space Inversion
	Appendix F: Proof of (2.21)
	Appendix G: Proof of (2.24) and Probabilistic Insights
	Appendix H: Another Proof of (2.24)
	Appendix I: Proof of (2.29) (Based on Fourier Methods)
	Appendix J: Another Proof of (2.29) (Based on Extension Methods)
	Appendix K: Proof of (2.36)
	Appendix L: Proof of (2.38)
	Appendix M: Another Proof of (2.38) (Based on (2.29))
	Appendix N: Proof of (2.46)
	Appendix O: Proof of (2.48)
	Appendix P: Proof of (2.52)
	Appendix Q: Proof of (2.53)
	Appendix R: Proof of (2.54)
	Appendix S: Proof of (2.60)
	Appendix T: Proof of (2.61)
	Appendix U: Proof of (2.62)
	Appendix V: Memory Effects of Caputo Type
	Appendix W: Proof of (3.7)
	Appendix X: Proof of (3.12)
	References
Dirichlet Problems for Fully Nonlinear Equations with ``Subquadratic\'\' Hamiltonians
	1 Content of the Paper
	2 Lipschitz Estimates
	3 Existence and Uniqueness Results for Homogenous Dirichlet Conditions
	4 Non Homogeneous Boundary Conditions
	References
Monotonicity Formulas for Static Metrics with Non-zero Cosmological Constant
	1 Introduction
		1.1 Static Einstein System
		1.2 Setting of the Problem and Statement of the Main Results (Case > 0)
		1.3 Setting of the Problem and Statement of the Main Results (Case < 0)
		1.4 Strategy of the Proof
		1.5 Summary
		1.6 Added Note
	2 Consequences
		2.1 Consequences on a Generic Level Set of u (Case > 0)
		2.2 The Geometry of M (Case > 0)
		2.3 Consequences on a Generic Level Set of u (Case < 0)
		2.4 The Geometry of M (Case < 0)
	3 A Conformally Equivalent Formulation of the Problem
		3.1 A Conformal Change of Metric (Case > 0)
		3.2 A Conformal Change of Metric (Case < 0)
		3.3 A Unifying Formalism
		3.4 The Geometry of the Level Sets of φ
		3.5 A Conformal Version of the Monotonicity-Rigidity Theorem
	4 Proof of Theorems 1.1 and 1.4 After Theorem 3.2
		4.1 Case > 0: Theorem 3.2 Implies Theorem 1.1
		4.2 Case < 0: Theorem 3.2 Implies Theorem 1.4
	5 Integral Identities
		5.1 First Integral Identity
		5.2 Second Integral Identity
	6 Proof of Theorem 3.2
		6.1 Continuity
		6.2 Monotonicity of 1(s)
		6.3 Differentiability
		6.4 The Second Derivative
	Appendix A: Technical Results
	Appendix B: Boucher-Gibbons-Horowitz Method
	References
Introduction to Controllability of Nonlinear Systems
	1 Control Systems as Families of Vector Fields
		1.1 Vec(Rn) and Its Lie Algebra
		1.2 Affine Control Systems
	2 The Krener Theorem: Local Accessibility
	3 Symmetric Systems
	4 Compatible Vector Fields
		4.1 Affine Systems with Recurrent Drift
		4.2 Affine Systems with Non-recurrent Drift
		4.3 Convexification
	5 Orbits and Necessary Conditions for Controllability
	References
Introduction to Variational Methods for Viscous Ergodic Mean-Field Games with Local Coupling
	1 Introduction
	2 Standing Assumptions
	3 Steady States of the Fokker-Planck Equation
	4 Hamilton Jacobi Equations with Coercive Hamiltonian
	5 Existence of Solutions to the MFG System
	6 Uniqueness of Solutions to the MFG System
	References
Flatness Results for Nonlocal Phase Transitions
	1 Introduction
	2 -Convergence Results for Nonlocal Phase Transitions
	3 The De Giorgi Conjecture for the Fractional Laplacian
		3.1 The Caffarelli-Silvestre Extension and the Notion of Minimality
		3.2 Stability of Solutions
		3.3 A Liouville-Type Result
		3.4 Sketch of the Proof of the De Giorgi Conjecture in Low Dimensions
		3.5 Energy Estimates
	4 Classification for Nonlocal Minimal Surfaces
		4.1 The Classical Setting
		4.2 The Nonlocal Setting
	5 What About Stable Objects?
		5.1 The Classical Setting
		5.2 The Nonlocal Setting
	References
Fractional De Giorgi Classes and Applications to Nonlocal Regularity Theory
	1 Introduction
	2 Fractional De Giorgi Classes
	3 Applications to Minimizers of Nonlocal Functionals
	4 Applications to Solutions of Nonlocal Equations
	Appendix A: An Explicit Example
	References
Harnack and Pointwise Estimates for Degenerate or Singular Parabolic Equations
	1 Introduction
	2 Elliptic Harnack Inequality
		2.1 Original Harnack
		2.2 Modern Developments
		2.3 Moser\'s Proof and Weak Harnack Inequalities
		2.4 Harnack Inequality on Minimal Surfaces
		2.5 Differential Harnack Inequality
		2.6 Beyond Smooth Manifolds
	3 Parabolic Harnack Inequality
		3.1 Original Parabolic Harnack
		3.2 The Linear Case with Coefficients
		3.3 First Consequences
		3.4 Riemannian Manifolds and Beyond
		3.5 The Nonlinear Setting
	4 Singular and Degenerate Parabolic Equations
		4.1 The Prototype Equation
		4.2 Regularity
		4.3 Intrinsic Harnack Inequalities
		4.4 Liouville Theorems
		4.5 Harnack Estimates at Large
	5 The Expansion of Positivity Approach
		5.1 Elliptic Equations
		5.2 Homogeneous Parabolic Equations
		5.3 Inhomogeneous Parabolic Equations
		5.4 Degenerate Parabolic Equations
		5.5 Singular Parabolic Equations
	References
Lectures on Curvature Flow of Networks
	1 Introduction
	2 Notation and Setting of the Problem
		2.1 Curves and Networks
		2.2 The Evolution Problem
			2.2.1 Formal Derivation of the Gradient Flow
			2.2.2 Geometric Problem
			2.2.3 The System of Quasilinear PDEs
	3 Short Time Existence and Uniqueness
		3.1 Existence and Uniqueness for the Special Flow
		3.2 Existence and Uniqueness
		3.3 Geometric Properties of the Flow
	4 Self-similar Solutions
	5 Integral Estimates
		5.1 Evolution of Length and Volume
		5.2 Evolution of the Curvature and Its Derivatives
		5.3 Consequences of the Estimates
	6 Analysis of Singularities
		6.1 Huisken\'s Monotonicity Formula
		6.2 Dynamical Rescaling
		6.3 Blow-Up Limits
	References
Maximum Principles at Infinity and the Ahlfors-Khas\'minskii Duality: An Overview
	1 Prelude: Maximum Principles at Infinity
		1.1 An Example: Immersions into Cones
		1.2 Parabolicity, Capacity and Evans Potentials
		1.3 Link with Stochastic Processes
		1.4 On Weak Formulations: The Case of Quasilinear Operators
	2 The General Framework
	3 Ahlfors, Khas\'minskii Properties and the AK-Duality
	4 Applications
		4.1 Completeness, Viscosity Ekeland Principle and ∞-Parabolicity
		4.2 The Hessian Principle and Martingale Completeness
		4.3 Laplacian Principles
	5 Partial Trace (Grassmannian) Operators
	6 AK-Duality and Polar Sets
	References
Singularities in the Calculus of Variations
	1 Introduction
	2 Preliminaries
	3 Linear Estimates and Consequences
		3.1 Energy Estimate
		3.2 Consequences for Minimizers
		3.3 Scalar Case
	4 Singular Examples
		4.1 Linear Elliptic Examples
		4.2 Rigidity Result for Homogeneity -n-22
		4.3 Null Lagrangian Approach of Šverák-Yan
	5 Parabolic Case
		5.1 Linear Estimates
		5.2 Singularities from Smooth Data
	References
Comparison Among Several Planar Fisher-KPP Road-Field Systems
	1 Introduction
	2 Preliminary Results: Comparison Principles, Long-Time Behavior and Existence of the Asymptotic Speed of Propagation
	3 Characterization of the Asymptotic Speed of Propagation for Problems with One Road
	4 Characterization of the Asymptotic Speed of Propagation for Problems with Two Roads
	References