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

 dc.contributor.advisor Fazly, Mostafa dc.contributor.advisor Gui, Changfeng dc.contributor.author Dimler, Bryan dc.contributor.committeeMember Popescu, Gelu dc.contributor.committeeMember Chen, Fengxin dc.creator.orcid https://orcid.org/0000-0002-9104-4951 dc.date.accessioned 2024-02-09T20:51:05Z dc.date.available 2024-02-09T20:51:05Z dc.date.issued 2020 dc.description.abstract We 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 ℝn and (−Δ)su = f(u) in ℝn, where (−Δ)s denotes the fractional Laplace operator with fractional exponent s ∈ (0,1). Here, we assume that f : ℝ → ℝ is at least of class C1. We begin by defining the fractional Laplace operator, and prove many of its fundamental properties. We then present the extension problem in ℝ+n+1 := {(x,y) ∈ ℝn+1 : x ∈ ℝn, y > 0} for the operator (−Δ)s introduced by Caffarelli and Silvestre (2007) and develop a fundamental solution and Poisson kernel for (−Δ)s in ℝ+n+1. 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)s, 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.department Mathematics dc.format.extent 155 pages dc.format.mimetype application/pdf dc.identifier.isbn 9798641000909 dc.identifier.uri https://hdl.handle.net/20.500.12588/3440 dc.language en dc.subject Allen-Cahn dc.subject calculus of variations dc.subject De Giorgi dc.subject fractional Laplacian dc.subject nonlocal dc.subject partial differential equations dc.subject.classification Mathematics dc.title De Giorgi's Conjecture for the Allen-Cahn Equation and Related Problems for Classical and Fractional Laplacians dc.type Thesis dc.type.dcmi Text dcterms.accessRights pq_closed thesis.degree.department Mathematics thesis.degree.grantor University of Texas at San Antonio thesis.degree.level Masters thesis.degree.name Master of Science

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