Geometric Measure Theory and Free Boundary Problems: Cetraro, Italy 2019 (Lecture Notes in Mathematics)

Contents
1 Introduction
	1.1 A-Free Measures and Applications
	1.2 Existence of Min-Max Minimal Hypersurfaces
	1.3 Regularity of Free Boundaries in Obstacle Problems
	1.4 Bernoulli Type Free Boundary Problems and Water Waves
2 A-Free Measures and Applications
	2.1 Introduction
		2.1.1 Example of Differential Operators
		2.1.2 Motivation
			Differential Complexes
			Non-linear PDE as Linear PDE Coupled with a Pointwise Constraint
			Inner Variation and the Stress Energy Tensor
			Stationary Varifolds
	2.2 Oscillation
	2.3 Linear PDE Constrained Measures
		2.3.1 Dimensional Estimates
	2.4 Elliptic Operators and Harmonic Analysis Tools
	Appendix A: A Brief Course on Radon Measures
		A.1 Basic Facts
		A.2 Tangent Measures
	References
3 Regularity of Free Boundaries in Obstacle Problems
	3.1 Introduction
		3.1.1 The Obstacle Problem
		3.1.2 On the Euler–Lagrange Equations
		3.1.3 The Free Boundary
	3.2 Some Motivations and Applications
		3.2.1 Fluid Filtration
		3.2.2 Phase Transitions
		3.2.3 Hele-Shaw Flow
		3.2.4 Optimal Stopping, Finance
		3.2.5 Interacting Particle Systems
	3.3 Basic Properties of Solutions
		3.3.1 Existence of Solutions
		3.3.2 Optimal Regularity of Solutions
		3.3.3 Nondegeneracy
		3.3.4 Summary of Basic Properties
	3.4 Regularity of Free Boundaries: An Overview
		3.4.1 Regularity of Free Boundaries: Main Results
		3.4.2 Overview of the Strategy
	3.5 Classification of Blow-Ups
		3.5.1 Convexity of Blow-Ups
		3.5.2 Homogeneity of Blow-Ups
		3.5.3 Classification of Blow-Ups
	3.6 Regularity of the Free Boundary
		3.6.1 C1,α Regularity of the Free Boundary
		3.6.2 Higher Regularity of the Free Boundary
	3.7 Singular Points
	3.8 The Size of the Singular Set
	3.9 Finer Understanding of Singular Points
		3.9.1 The Thin Obstacle Problem
			The Free Boundary in the Thin Obstacle Problem
			Dimension-Reduction in the Thin Obstacle Problem
		3.9.2 Relating the Obstacle Problem and the Thin Obstacle Problem
	Appendix: Proof of Proposition 12
	References
4 Bernoulli Type Free Boundary Problems and Water Waves
	4.1 Introduction
	4.2 Relation to Water Waves
	4.3 Background Knowledge on the Bernoulli Problem
	4.4 Notation
	4.5 Notion of Solution
	4.6 Monotonicity Formula and Consequences
	4.7 The Two-dimensional Case
	4.8 Definition of Degenerate Points
	4.9 Frequency Formula
	4.10 Blow-Up Limits
	4.11 Concentration Compactness in Two Dimensions
	4.12 Degenerate Points in Two Dimensions
	4.13 Main Reuslt
	4.14 Extensions
	References