De Giorgi's Conjecture for the Allen-Cahn Equation and Related Problems for Classical and Fractional Laplacians

dc.contributor.advisorFazly, Mostafa
dc.contributor.advisorGui, Changfeng
dc.contributor.authorDimler, Bryan
dc.contributor.committeeMemberPopescu, Gelu
dc.contributor.committeeMemberChen, Fengxin
dc.creator.orcidhttps://orcid.org/0000-0002-9104-4951
dc.date.accessioned2024-02-09T20:51:05Z
dc.date.available2024-02-09T20:51:05Z
dc.date.issued2020
dc.description.abstractWe present results related to a famous conjecture of Enrico De Giorgi for a special class of bounded monotone solutions, called layer solutions, to nonlinear equations of the form −Δu = f(u) in ℝ<sup>n</sup> and (−Δ)<sup>s</sup>u = f(u) in ℝ<sup>n</sup>, where (−Δ)<sup>s</sup> denotes the fractional Laplace operator with fractional exponent s ∈ (0,1). Here, we assume that f : ℝ → ℝ is at least of class C<sup>1</sup>. We begin by defining the fractional Laplace operator, and prove many of its fundamental properties. We then present the extension problem in ℝ<sub>+</sub><sup>n+1</sup> := {(x,y) ∈ ℝ<sup>n+1</sup> : x ∈ ℝ<sup>n</sup>, y > 0} for the operator (−Δ)<sup>s</sup> introduced by Caffarelli and Silvestre (2007) and develop a fundamental solution and Poisson kernel for (−Δ)<sup>s</sup> in ℝ<sub>+</sub><sup>n+1</sup>. Subsequently, we prove De Giorgi's conjecture for layer solutions to the first equation above in dimensions n ≤ 3. Precisely, we show that layer solutions are necessarily one-dimensional. We then turn our attention to the fractional De Giorgi conjecture, and present the fractional versions of the results obtained in the classical case (i.e. similar results for the second equation above). To supplement, we discuss some Pohozaev-type monotonicity formulae for the operators Δ and (−Δ)<sup>s</sup>, along with some closely related problems. We close with a brief discussion of some open problems and topics of further interest to the author.
dc.description.departmentMathematics
dc.format.extent155 pages
dc.format.mimetypeapplication/pdf
dc.identifier.isbn9798641000909
dc.identifier.urihttps://hdl.handle.net/20.500.12588/3440
dc.languageen
dc.subjectAllen-Cahn
dc.subjectcalculus of variations
dc.subjectDe Giorgi
dc.subjectfractional Laplacian
dc.subjectnonlocal
dc.subjectpartial differential equations
dc.subject.classificationMathematics
dc.titleDe Giorgi's Conjecture for the Allen-Cahn Equation and Related Problems for Classical and Fractional Laplacians
dc.typeThesis
dc.type.dcmiText
dcterms.accessRightspq_closed
thesis.degree.departmentMathematics
thesis.degree.grantorUniversity of Texas at San Antonio
thesis.degree.levelMasters
thesis.degree.nameMaster of Science

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