Optimal transport, free boundary regularity, and stability results for geometric and functional inequalities

dc.contributor.advisorFigalli, Alessioen
dc.contributor.committeeMemberCaffarelli, Luis Aen
dc.contributor.committeeMemberGamba, Irene Men
dc.contributor.committeeMemberBeckner, Williamen
dc.contributor.committeeMemberKoch, Hansen
dc.contributor.committeeMemberRoquejoffre, Jean-Michelen
dc.creatorIndrei, Emanuel Gabrielen
dc.date.accessioned2013-07-01T19:50:44Zen
dc.date.issued2013-05en
dc.date.submittedMay 2013en
dc.date.updated2013-07-01T19:50:45Zen
dc.descriptiontexten
dc.description.abstractWe investigate stability for certain geometric and functional inequalities and address the regularity of the free boundary for a problem arising in optimal transport theory. More specifically, stability estimates are obtained for the relative isoperimetric inequality inside convex cones and the Gaussian log-Sobolev inequality for a two parameter family of functions. Thereafter, away from a ``small" singular set, local C^{1,\alpha} regularity of the free boundary is achieved in the optimal partial transport problem. Furthermore, a technique is developed and implemented for estimating the Hausdorff dimension of the singular set. We conclude with a corresponding regularity theory on Riemannian manifolds.en
dc.description.departmentMathematics
dc.format.mimetypeapplication/pdfen
dc.identifier.urihttp://hdl.handle.net/2152/20631en
dc.language.isoen_USen
dc.subjectOptimal transport theoryen
dc.subjectthe relative isoperimetric inequalityen
dc.subjectthe log-Sobolev inequalityen
dc.subjectfree boundary regularityen
dc.subjectpartial differential equationsen
dc.subjectquantitative stabilityen
dc.subjectthe optimal partial transport problem.en
dc.titleOptimal transport, free boundary regularity, and stability results for geometric and functional inequalitiesen
thesis.degree.departmentMathematicsen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorThe University of Texas at Austinen
thesis.degree.levelDoctoralen
thesis.degree.nameDoctor of Philosophyen

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