Free boundary problems for harmonic and caloric measure /
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| Author / Creator: | Engelstein, Max, author. |
|---|---|
| Imprint: | 2016. Ann Arbor : ProQuest Dissertations & Theses, 2016 |
| Description: | 1 electronic resource (232 pages) |
| Language: | English |
| Format: | E-Resource Dissertations |
| Local Note: | School code: 0330 |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/10862849 |
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| 100 | 1 | |a Engelstein, Max, |e author. | |
| 245 | 1 | 0 | |a Free boundary problems for harmonic and caloric measure / |c Engelstein, Max David. |
| 260 | |c 2016. | ||
| 264 | 1 | |a Ann Arbor : |b ProQuest Dissertations & Theses, |c 2016 | |
| 300 | |a 1 electronic resource (232 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 500 | |a Advisors: Carlos E. Kenig Committee members: Wilhelm Schlag. | ||
| 502 | |b Ph.D. |c University of Chicago, Division of the Physical Sciences, Department of Mathematics |d 2016. | ||
| 510 | 4 | |a Dissertation Abstracts International, |c Volume: 77-10(E), Section: B. | |
| 520 | |a In this paper we consider two free boundary problems, which we solve using a combination of techniques and tools from harmonic analysis, geometric measure theory and partial differential equations. The first problem is a two-phase problem for harmonic measure, initially studied by Kenig and Toro [KT06]. The central difficulty in that problem is the possibility of degeneracy; losing geometric information at a point where both phases vanish. We establish non-degeneracy by proving that the Almgren frequency formula, applied to an appropriately constructed function, is "almost monotone". In this way, we prove a sharp Holder regularity result (this work was originally published in [Eng14]). | ||
| 520 | |a The second problem is a one-phase problem for caloric measure, initially posed by Hofmann, Lewis and Nystrom [HLN04]. Here the major difficulty is to classify the "flat blowups". We do this by adapting work of Andersson and Weiss [AW09], who analyzed a related problem arising in combustion. This classification allows us to generalize results of [KT03] to the parabolic setting and answer in the affirmative a question left open in the aforementioned paper of Hofmann et al. (this work was originally published in [Eng15]). | ||
| 546 | |a English | ||
| 590 | |a School code: 0330 | ||
| 690 | |a Mathematics. | ||
| 710 | 2 | |a University of Chicago. |e degree granting institution. | |
| 720 | 1 | |a Carlos E. Kenig |e degree supervisor. | |
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