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

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References

SHOWING 1-10 OF 28 REFERENCES
A Harnack Inequality Approach to the Regularity of Free Boundaries. Part I: Lipschitz Free Boundaries are $C^{1, \alpha}$
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. WeExpand
A gradient bound for free boundary graphs
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 freeExpand
Some remarks on stability of cones for the one-phase free boundary problem
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
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 spanningExpand
An overdetermined problem in potential theory
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 whichExpand
A Geometric Approach to Free Boundary Problems
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 boundariesExpand
The space of embedded minimal surfaces of fixed genus in a 3-manifold III; Planar domains
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 whereExpand
A survey on classical minimal surface theory
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 theExpand
The space of embedded minimal surfaces of fixed genus in a 3-manifold II; Multi-valued graphs in disks
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 understandingExpand
The space of embedded minimal surfaces of fixed genus in a 3-manifold IV; Locally simply connected
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 ofExpand
...
1
2
3
...