Free Boundary Problem

2025 Cambridge Mathematics Master's dissertation overview on the Alt-Caffarelli one-phase free boundary problem, low-dimensional regularity, and stability methods.

This repository provides a concise public-facing overview of my dissertation on the Alt-Caffarelli one-phase free boundary problem. The full dissertation was written in LaTeX and spans 45 pages. This repository is intended as a readable research summary rather than a complete reproduction of the dissertation.

Overview

The dissertation studies the one-phase free boundary problem associated with the Alt-Caffarelli functional

$$J(v, \Omega) = \int_{\Omega} \left( |\nabla v|^2 + \chi_{{v > 0}} Q^2 \right).$$

Minimizers formally satisfy the Euler-Lagrange conditions

$$u \geq 0, \qquad \Delta u = 0 \quad \text{in } \Omega \cap {u > 0}, \qquad |\nabla u| = Q \quad \text{weakly on } \Omega \cap \partial {u > 0}.$$

The boundary condition is imposed on a set determined by the solution itself. This gives rise to a free boundary problem.

The main focus of the dissertation is the classification of global minimizers in low dimensions and its connection to the regularity of their free boundaries.

Main Result Studied

The dissertation works towards the following regularity theorem.

In dimensions $n \leq 4$, the free boundary of global minimizers to the Alt-Caffarelli functional is locally a $C^{1,\alpha}$-hypersurface. If $Q$ is smooth, then the free boundary is a smooth hypersurface.

This result is connected to the classification of homogeneous global minimizers and the analysis of possible singular free boundaries.

The higher-dimensional classification problem remains open. In particular, the critical dimension $k^*$ is known to satisfy

$$k^* \in {5, 6, 7}.$$

Mathematical Context

Free boundary problems arise when the domain on which a PDE is solved is itself unknown and must be determined as part of the solution.

In this problem, the positivity set

$${u > 0}$$

is not fixed in advance. Its boundary,

$$\partial {u > 0},$$

is the free boundary.

The central regularity question is whether this free boundary is smooth, singular, or somewhere in between.

This dissertation sits at the intersection of:

• Calculus of variations
• Elliptic partial differential equations
• Geometric measure theory
• Free boundary regularity
• Blow-up analysis
• Stability methods
• Minimal surface theory analogues

Dissertation Structure

The dissertation follows a roughly chronological development of the theory.

Section	Content
Section 1	Problem setup and existence of minimizers
Section 2	Local minima and Euler-Lagrange conditions
Section 3	Blow-up sequences and blow-up limits
Section 4	Weiss' monotonicity formula and dimension reduction
Section 5	Stability and the Caffarelli-Jerison-Kenig inequality
Section 6	Jerison-Savin instability criterion and low-dimensional regularity
Final part	Reformulation of the subsolution construction approach as an optimization problem

Key Ideas

1. Variational Formulation

The problem begins with the minimization of the functional

$$J(v, \Omega) = \int_{\Omega} \left( |\nabla v|^2 + \chi_{{v > 0}} Q^2 \right).$$

The Dirichlet energy term $|\nabla v|^2$ penalizes oscillation, while the characteristic term $\chi_{{v > 0}} Q^2$ penalizes the size of the positivity set.

The competition between these two terms allows minimizers to develop regions where they vanish identically. The boundary of such a region is not prescribed in advance, which is why the problem has a free boundary.

2. Existence and Local Minima

The dissertation begins by reviewing the variational setup introduced by Alt and Caffarelli.

Using Hilbert's direct method, together with compactness and coercivity tools such as the Rellich-Kondrachov compactness theorem and generalized Poincare-type inequalities, one obtains the existence of minimizers.

The next step is to study local minimizers and derive the Euler-Lagrange conditions they satisfy. These conditions include harmonicity in the positivity set and a Bernoulli-type condition on the free boundary.

3. Lipschitz Continuity and Non-Degeneracy

A major early part of the theory is to extract regularity information from the minimality of the functional.

Two important properties are:

• Lipschitz continuity, which gives upper control on how quickly a minimizer can grow
• Non-degeneracy, which prevents a minimizer from vanishing too slowly near the free boundary

Together, these properties provide the compactness and growth control needed for blow-up analysis.

4. Blow-Up Analysis

To understand the local structure of the free boundary near a point $x_0$, one studies rescaled functions of the form

$$u_r(x) = \frac{u(x_0 + rx)}{r}.$$

As $r \to 0$, subsequential limits of $u_r$ are called blow-up limits.

These blow-up limits capture the infinitesimal geometry of the free boundary near $x_0$. Classifying possible blow-up limits is therefore central to understanding the regularity of the free boundary itself.

5. Weiss' Monotonicity Formula

Weiss' monotonicity formula is used to show that blow-up limits are homogeneous of degree one.

That is, a blow-up limit $u_0$ satisfies

$$u_0(\lambda x) = \lambda u_0(x) \qquad \text{for all } \lambda > 0.$$

This reduces the local regularity problem to the classification of homogeneous global minimizers.

6. Dimension Reduction

The dissertation introduces the critical dimension $k^*$, which is the lowest dimension in which singular minimizing cones may exist.

Weiss' dimension reduction technique shows why low-dimensional classification results are powerful. If singular minimizers cannot occur in low dimensions, then the free boundary must be regular in those dimensions.

The classification in dimensions $n \leq 4$ implies regularity of the free boundary in those dimensions.

7. Stability and Instability

A key part of the dissertation studies stability of homogeneous degree-one solutions for the modified Weiss problem.

The Caffarelli-Jerison-Kenig integral inequality provides a stability condition. The Jerison-Savin approach then gives an instability criterion through the construction of appropriate subsolutions.

This framework is used to analyze the relevant homogeneous solutions in dimensions

$$n = 3 \qquad \text{and} \qquad n = 4.$$

The resulting lower bound on the critical dimension is

$$k^* \geq 5.$$

8. Optimization Reformulation

The final part of the dissertation reformulates the subsolution construction approach as an optimization problem.

This perspective provides a different way to view the instability argument and connects the analytic construction of subsolutions with an optimization-based formulation.

Main Techniques

The dissertation reviews and reconstructs the following mathematical tools.

Technique	Purpose
Hilbert's direct method	Proves existence of minimizers
Rellich-Kondrachov compactness	Provides compactness for minimizing sequences
Generalized Poincare inequalities	Supports coercivity and compactness arguments
Euler-Lagrange analysis	Derives PDE and free boundary conditions
Lipschitz estimates	Controls growth of minimizers
Non-degeneracy estimates	Prevents excessive flattening near the free boundary
Blow-up analysis	Studies local free boundary geometry
Weiss' monotonicity formula	Establishes homogeneity of blow-up limits
Dimension reduction	Relates singularities to lower-dimensional cones
Stability inequalities	Characterizes stability of homogeneous solutions
Subsolution construction	Proves instability in selected dimensions
Optimization reformulation	Recasts part of the instability argument in optimization terms

Main References

The dissertation draws on foundational work in the theory of free boundary problems, including:

Authors	Contribution
Alt and Caffarelli	One-phase free boundary framework
Alt, Caffarelli, and Friedman	Two-phase free boundary theory and applications
Aguilera, Caffarelli, and Spruck	Variational formulations in heat conduction problems
Weiss	Monotonicity formula and dimension reduction methods
Caffarelli, Jerison, and Kenig	Stability inequalities for free boundary problems
Jerison and Savin	Instability criterion and low-dimensional regularity analysis
De Silva and Jerison	Singular energy-minimizing free boundaries

Repository Contents

This repository is a public-facing overview of my dissertation rather than a full reproduction of the 45-page LaTeX document.

README.md
notes/
  variational_equations.md

• README.md gives a high-level overview of the problem, main theorem, mathematical context, and techniques studied in the dissertation.
• notes/variational_equations.md gives a short technical taster from Section 2, showing how minimality of the Alt-Caffarelli functional leads to subharmonicity, harmonicity on the positivity set, and the weak free boundary condition.

The full dissertation is not included here.

Skills Demonstrated

This dissertation demonstrates experience with:

• Advanced mathematical exposition
• Variational methods
• Elliptic PDE theory
• Free boundary regularity
• Blow-up and compactness arguments
• Stability analysis
• Reading and synthesizing research papers
• Technical writing in LaTeX
• Reformulating mathematical arguments from an optimization perspective

Note

This repository is intended for academic and professional reference. It provides a concise overview of the dissertation's mathematical content without reproducing the full document.