# Structure of one-phase free boundaries in the plane

@article{Jerison2014StructureOO, title={Structure of one-phase free boundaries in the plane}, author={D. Jerison and Nikola Kamburov}, journal={arXiv: Analysis of PDEs}, year={2014} }

We study classical solutions to the one-phase free boundary problem in which the free boundary consists of smooth curves and the components of the positive phase are simply-connected. We show that if two components of the free boundary are close, then the solution locally resembles an entire solution discovered by Hauswirth, H\'elein and Pacard, whose free boundary has the shape of a double hairpin. Our results are analogous to theorems of Colding and Minicozzi characterizing embedded minimal… Expand

#### Figures from this paper

#### 14 Citations

Free boundaries subject to topological constraints

- Mathematics
- 2019

We discuss the extent to which solutions to one-phase free boundary problems can be characterized according to their topological complexity. Our questions are motivated by fundamental work of Luis… Expand

On smooth solutions to one phase free boundary problem in R n

- 2017

We construct a smooth axially symmetric solution to the classical one phase free boundary problem in R, n ≥ 3. Its free boundary is of “catenoid” type. This is a higher dimensional analogy of the… Expand

On Smooth Solutions to One Phase-Free Boundary Problem in $\mathbb{R}^{n}$

- Mathematics
- International Mathematics Research Notices
- 2019

We construct a smooth axially symmetric solution to the classical one phase free boundary problem in $\mathbb{R}^{n}$, $n\geq 3.$ Its free boundary is of “catenoid” type. This is a higher… Expand

The structure of finite Morse index solutions to two free boundary problems in $\mathbb{R}^2$

- Mathematics
- 2015

We give a description of the structure of finite Morse index solutions to two free boundary problems in $\mathbb{R}^2$. These free boundary problems are models of phase transition and they are… Expand

On one phase free boundary problem in $\mathbb{R}^{N}$

- Mathematics
- 2017

We construct a smooth axially symmetric solution to the classical one phase free boundary problem in $\mathbb{R}^{N}$. Its free boundary is of \textquotedblleft catenoid\textquotedblright\ type. This… Expand

Higher Critical Points in an Elliptic Free Boundary Problem

- Mathematics
- 2017

We study higher critical points of the variational functional associated with a free boundary problem related to plasma confinement. Existence and regularity of minimizers in elliptic free boundary… Expand

On a free boundary problem and minimal surfaces

- Mathematics
- 2017

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov-Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two… Expand

Finite Morse index implies finite ends

- Physics, Mathematics
- 2017

We prove that finite Morse index solutions to the Allen-Cahn equation in R have finitely many ends and linear energy growth. The main tool is a curvature decay estimate on level sets of these finite… Expand

The one-phase problem for harmonic measure in two-sided NTA domains

- Mathematics
- 2016

We show that if $\Omega\subset\mathbb R^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\textrm{VMO}(\sigma)$, where $\sigma$ is the… Expand

On Nonminimizing Solutions of Elliptic Free Boundary Problems

- Mathematics
- 2019

We present a variational framework for studying the existence and regularity of solutions to elliptic free boundary problems that do not necessarily minimize energy. As applications, we obtain… Expand

#### References

SHOWING 1-10 OF 28 REFERENCES

A Harnack Inequality Approach to the Regularity of Free Boundaries. Part I: Lipschitz Free Boundaries are $C^{1, \alpha}$

- Mathematics
- 1987

This is the first in a series of papers where we intend to show, in several steps, the existence of classical (or as classical as possible) solutions to a general two-phase free-boundary system. We… Expand

A gradient bound for free boundary graphs

- Mathematics
- 2010

We prove an analogue for a one-phase free boundary problem of the classical gradient bound for solutions to the minimal surface equation. It follows, in particular, that every energy-minimizing free… Expand

Some remarks on stability of cones for the one-phase free boundary problem

- Mathematics
- 2014

We show that stable cones for the one-phase free boundary problem are hyperplanes in dimension 4. As a corollary, both one and two-phase energy minimizing hypersurfaces are smooth in dimension 4.

Global Properties of Minimal Surfaces in E 3 and E n

- Mathematics
- 1964

Abstract : A minimal surface is the surface of least area bounded by a given closed curve. In three dimensional space these surface are realized, for reasonably simple curves, by soap films spanning… Expand

An overdetermined problem in potential theory

- Mathematics
- 2013

We investigate a problem posed by L. Hauswirth, F. H\'elein, and F. Pacard, namely, to characterize all the domains in the plane that admit a "roof function", i.e., a positive harmonic function which… Expand

A Geometric Approach to Free Boundary Problems

- Mathematics
- 2005

Elliptic problems: An introductory problem Viscosity solutions and their asymptotic developments The regularity of the free boundary Lipschitz free boundaries are $C^{1,\gamma}$ Flat free boundaries… Expand

The space of embedded minimal surfaces of fixed genus in a 3-manifold III; Planar domains

- Physics, Mathematics
- 2002

This paper is the third in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. In [CM3]–[CM5] we describe the case where… Expand

A survey on classical minimal surface theory

- Mathematics
- 2012

Meeks and Perez present a survey of recent spectacular successes in classical minimal surface theory. The classification of minimal planar domains in three-dimensional Euclidean space provides the… Expand

The space of embedded minimal surfaces of fixed genus in a 3-manifold II; Multi-valued graphs in disks

- Mathematics
- 2002

This paper is the second in a series where we give a description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key for understanding… Expand

The space of embedded minimal surfaces of fixed genus in a 3-manifold IV; Locally simply connected

- Mathematics
- 2002

This paper is the fourth in a series where we describe the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3manifold. The key is to understand the structure of… Expand