Regularity of elliptic transmission problems and a new family of integro-differential operators related to the Monge-Ampère equation

dc.contributor.advisorCaffarelli, Luis A.
dc.contributor.advisorStinga, Pablo Raúl
dc.contributor.committeeMemberGamba, Irene M
dc.contributor.committeeMemberPatrizi, Stefania
dc.contributor.committeeMemberVasseur, Alexis F
dc.creatorSoria-Carro, María
dc.creator.orcid0000-0001-7127-7646
dc.date.accessioned2022-11-02T00:40:47Z
dc.date.available2022-11-02T00:40:47Z
dc.date.created2022-08
dc.date.issued2022-07-12
dc.date.submittedAugust 2022
dc.date.updated2022-11-02T00:40:49Z
dc.description.abstractThis dissertation is divided into two main topics. First, we study transmission problems for elliptic equations, both linear and nonlinear, and prove existence, uniqueness, and optimal regularity of solutions. In our first work, we consider a problem for harmonic functions and use geometric techniques. Our second work considers viscosity solutions to fully nonlinear transmission problems. Given the nonlinear nature of these equations, our arguments are based on perturbation methods and comparison principles. The second topic is related to nonlocal Monge-Ampère equations. We define a new family of integro-differential equations arising from geometric considerations and study some of their properties. Furthermore, we consider a Poisson problem in the full space and prove existence, uniqueness, and C¹,¹ regularity of solutions. For this problem, we use tools from convex analysis and symmetrization.
dc.description.departmentMathematics
dc.format.mimetypeapplication/pdf
dc.identifier.urihttps://hdl.handle.net/2152/116448
dc.identifier.urihttp://dx.doi.org/10.26153/tsw/43343
dc.language.isoen
dc.subjectTransmission problems
dc.subjectElliptic regularity
dc.subjectFully nonlinear equations
dc.subjectViscosity solutions
dc.subjectIntegro-differential operators
dc.subjectMonge-Ampère
dc.subjectRearrangements
dc.titleRegularity of elliptic transmission problems and a new family of integro-differential operators related to the Monge-Ampère equation
dc.typeThesis
dc.type.materialtext
thesis.degree.departmentMathematics
thesis.degree.disciplineMathematics
thesis.degree.grantorThe University of Texas at Austin
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy

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