Optimal regularity and nondegeneracy for minimizers of an energy related to the fractional Laplacian
| dc.contributor.advisor | Caffarelli, Luis A. | en |
| dc.contributor.committeeMember | Arapostathis, Ari | en |
| dc.contributor.committeeMember | Beckner, William | en |
| dc.contributor.committeeMember | Tsai, Richard | en |
| dc.contributor.committeeMember | Vasseur, Alexis | en |
| dc.creator | Yang, Ray | en |
| dc.date.accessioned | 2011-10-25T17:18:29Z | en |
| dc.date.available | 2011-10-25T17:18:29Z | en |
| dc.date.issued | 2011-08 | en |
| dc.date.submitted | August 2011 | en |
| dc.date.updated | 2011-10-25T17:18:35Z | en |
| dc.description | text | en |
| dc.description.abstract | We study the optimal regularity and nondegeneracy of a free boundary problem related to the fractional Laplacian through the extension technique of Caffarelli and Silvestre. Specifically, we show that minimizers of the energy [mathematical equation] where [mathematical equations] with 0 < [gamma] < 1, with free behavior on the set {y=0}, are Holder continuous with exponent [Beta] = 2[sigma]/2-[gamma]. These minimizers exhibit a free boundary: along {y = 0}, they divide into a zero set {u = 0} and a positivity set where {u > 0}; we call the interface between these sets the free boundary. The regularity is optimal, due to the non-degeneracy property of the minimizers: in any ball of radius r centered at the free boundary, the minimizer grows (in the supremum sense) like r[Beta]. This work is related to, but addresses a different problem from, recent work of Caffarelli, Roquejoffre, and Sire. | en |
| dc.description.department | Mathematics | en |
| dc.description.department | en | |
| dc.format.mimetype | application/pdf | en |
| dc.identifier.slug | 2152/ETD-UT-2011-08-3926 | en |
| dc.identifier.uri | http://hdl.handle.net/2152/ETD-UT-2011-08-3926 | en |
| dc.language.iso | eng | en |
| dc.subject | Free boundary | en |
| dc.subject | Optimal regularity | en |
| dc.subject | Fractional Laplacian | en |
| dc.title | Optimal regularity and nondegeneracy for minimizers of an energy related to the fractional Laplacian | en |
| dc.type.genre | thesis | en |
| thesis.degree.department | Mathematics | en |
| thesis.degree.discipline | Mathematics | en |
| thesis.degree.grantor | University of Texas at Austin | en |
| thesis.degree.level | Doctoral | en |
| thesis.degree.name | Doctor of Philosophy | en |