# Regularity theory for the symmetric minimal surface equation

## Abstract/Contents

- Abstract
- We consider the solutions of the symmetric minimal surface equation (SME), the geometric significance of which is that such solutions have symmetric graphs, which are minimal hypersurfaces. The SME has been previously studied by Dierkes, Huisken, and Simon. Using the uniform Holder continuity estimate for positive bounded solutions proved by Simon, we are able to define a singular solution of the SME as local uniform limits of sequences of positive solutions of the SME. Next, we study the existence of singular solutions, beginning with the observation that, by standard ODE theory, one-dimensional singular solutions do not exist. However we will show, via a Leray-Schauder argument, that in all dimensions greater than one there is rich class of singular solutions. Having established an existence theory for the SME, we focus on studying the regularity and singularity properties of singular solutions of the SME. More or less standard quasilinear elliptic theory shows that for any bounded positive solution of the SME there are local gradient bounds, and for positive solutions defined on the closure of a smooth convex domain there is a global gradient bound, depending only on the supremum and infimum of the solution and the domain. For singular solutions (or classes of positive solutions without a uniform positive lower bound), it will be shown that there is a local gradient estimate in a neighborhood of any singular point, depending only on the supremum of the solution and the distance of the singular point and the boundary of the domain. To establish these latter gradient estimates, we will use a modification of Simon's blow up of the Jacobi field argument together with the work of Ilmanen. As a consequence of gradient estimates, we are then able to show, via an application of the Federer dimension reducing argument, that the Hausdorff dimension of the singular set of an n-dimensional SME is at most n-2.

## Description

Type of resource | text |
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Form | electronic; electronic resource; remote |

Extent | 1 online resource. |

Publication date | 2012 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Associated with | Fouladgar, Kaveh |
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Associated with | Stanford University, Department of Mathematics |

Primary advisor | Brendle, Simon, 1981- |

Primary advisor | Simon, L. (Leon), 1945- |

Thesis advisor | Brendle, Simon, 1981- |

Thesis advisor | Simon, L. (Leon), 1945- |

Thesis advisor | Schoen, Richard (Richard M.) |

Thesis advisor | White, Brian, 1957- |

Advisor | Schoen, Richard (Richard M.) |

Advisor | White, Brian, 1957- |

## Subjects

Genre | Theses |
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## Bibliographic information

Statement of responsibility | Kaveh Fouladgar. |
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Note | Submitted to the Department of Mathematics. |

Thesis | Thesis (Ph.D.)--Stanford University, 2012. |

Location | electronic resource |

## Access conditions

- Copyright
- © 2012 by Kaveh Fouladgar
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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