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dc.contributor.advisorLlave, Rafael de laen
dc.creatorBlass, Timothy Jamesen
dc.date.accessioned2011-06-01T13:51:55Zen
dc.date.accessioned2011-06-01T13:52:06Zen
dc.date.available2011-06-01T13:51:55Zen
dc.date.available2011-06-01T13:52:06Zen
dc.date.issued2011-05en
dc.date.submittedMay 2011en
dc.identifier.urihttp://hdl.handle.net/2152/ETD-UT-2011-05-2798en
dc.descriptiontexten
dc.description.abstractThis dissertation is organized into four chapters: an introduction followed by three chapters, each based on one of three separate papers. In Chapter 2 we consider gradient descent equations for energy functionals of the type [mathematical equation] where A is a second-order uniformly elliptic operator with smooth coefficients. We consider the gradient descent equation for S, where the gradient is an element of the Sobolev space H[superscipt beta], [beta is an element of](0, 1), with a metric that depends on A and a positive number [gamma] > sup |V₂₂|. The main result of Chapter 2 is a weak comparison principle for such a gradient flow. We extend our methods to the case where A is a fractional power of an elliptic operator, and we provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional. In Chapter 3 we investigate the differentiability of the minimal average energy associated to the functionals [mathematical equation] using numerical and perturbation methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the minimal average energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the perturbation parameter [epsilon], and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series. In Chapter 4 we present a method for determining the stability of a class of stochastically forced ordinary differential equations, where the forcing term can be obtained by passing white noise through a filter of arbitrarily high degree. We use the Fokker-Planck equation to write a partial differential equation for the second moments, which we turn into an eigenvalue problem for a second-order differential operator. We develop ladder operators to determine analytic expressions for the eigenvalues and eigenfunctions of this differential operator, and thus determine the stability.en
dc.format.mimetypeapplication/pdfen
dc.language.isoengen
dc.subjectPartial differential equationsen
dc.subjectAubry-Mather theoryen
dc.subjectComparison principleen
dc.subjectLindstedt seriesen
dc.subjectCell problemen
dc.subjectStabilityen
dc.subjectStochastically forced ordinary differential equationsen
dc.titleOn the Aubry-Mather theory for partial differential equations and the stability of stochastically forced ordinary differential equationsen
dc.date.updated2011-06-01T13:52:06Zen
dc.contributor.committeeMemberCaffarelli, Luisen
dc.contributor.committeeMemberKoch, Hansen
dc.contributor.committeeMemberRadin, Charlesen
dc.contributor.committeeMemberRodin, Gregen
dc.contributor.committeeMemberYing, Lexingen
dc.description.departmentMathematicsen
dc.type.genrethesisen
thesis.degree.departmentMathematicsen
thesis.degree.disciplineMathematicsen
thesis.degree.grantorUniversity of Texas at Austinen
thesis.degree.levelDoctoralen
thesis.degree.nameDoctor of Philosophyen


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