Geometric variational problems : regular and singular behavior
Abstract/Contents
- Abstract
- This thesis is devoted to the study of two of the most fundamental geometric variational problems, namely the harmonic map problem and the minimal surface problem. Chapter 1 concerns the partial regularity of energy-minimizing harmonic maps between Riemannian manifolds. In the case where both the domain and the target metrics are smooth, the regularity theory is rather well-developed, and we managed to extend part of this theory to the case where the domain metric is only bounded measurable. Specifically, we show that in this case the singular set of an energy-minimizing map always has codimension strictly larger than two. In Chapter 2, we study the existence of codimension-two minimal submanifolds in a closed Riemannian manifold using the phase-transition approach, which is an alternative to the classical min-max theory and has enjoyed great success in the codimesion-one case. To be precise, we show that one can obtain a codimension-two stationary rectifiable varifold as the energy concentration set of a sequence of suitably bounded critical points of the Ginzburg-Landau functional, which was originally a model for phase transition phenomena in superconducting materials. In Appendix A, we consider the fundamental solution of second-order elliptic systems in divergence form, and prove that under mild assumptions on the coefficients, the fundamental solution can be bounded from above by the Green's function for the Laplacian. Such a growth estimate plays an important role in the analysis in Chapter 2, but is perhaps also interesting on its own.
Description
| Type of resource | text |
|---|---|
| Form | electronic; electronic resource; remote |
| Extent | 1 online resource. |
| Publication date | 2017 |
| Issuance | monographic |
| Language | English |
Creators/Contributors
| Associated with | Cheng, Da Rong |
|---|---|
| Associated with | Stanford University, Department of Mathematics. |
| Primary advisor | Schoen, Richard (Richard M.) |
| Primary advisor | White, Brian, 1957- |
| Thesis advisor | Schoen, Richard (Richard M.) |
| Thesis advisor | White, Brian, 1957- |
| Advisor | Hershkovits, Or |
Subjects
| Genre | Theses |
|---|
Bibliographic information
| Statement of responsibility | Da Rong Cheng. |
|---|---|
| Note | Submitted to the Department of Mathematics. |
| Thesis | Thesis (Ph.D.)--Stanford University, 2017. |
| Location | electronic resource |
Access conditions
- Copyright
- © 2017 by Da Rong Cheng
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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