Dissertation/Thesis Abstract

On the infinity Laplacian and Hrushovski's fusion
by Smart, Charles Krug, Ph.D., University of California, Berkeley, 2010, 68; 3413555
Abstract (Summary)

We study viscosity solutions of the partial differential equation [special characters omitted] where U ⊆ [special characters omitted] is bounded and open, fC( U) ∩ L (U), and [special characters omitted] is the infinity Laplacian.

Our first result is the Max-Ball Theorem, which states that if uUSC(U) is a viscosity subsolution of [special characters omitted] and ϵ > 0, then the function v( x) := max( x,ϵ) u satisfies [special characters omitted] for all xU := {x ∈ U : dist(x, ∂ U) > 2ϵ}. The left-hand side of this latter inequality is a monotone finite difference scheme that is comparatively easy to analyze. The Max-Ball Theorem allows us to lift results for this finite difference scheme to the continuum equation. In particular, we obtain a new proof of uniqueness of viscosity solutions to the Dirichlet problem when f ≡ 0, inf f > 0, or sup f < 0. The results mentioned so far are joint work with S. Armstrong.

The Max-Ball Theorem is also useful in the analysis of numerical methods for the infinity Laplacian. We obtain a rate of convergence for the numerical method of Oberman [32]. We also present a new adaptive finite difference scheme.

We also prove some results in Model Theory. We study rank-preserving interpretations of theories of finite Morley rank in strongly minimal sets. In particular, we partially answer a question posed by Hasson [20], showing that definable degree is not necessary for such interpretations. We generalize Ziegler's fusion of structures of finite Morley rank [38] to a class of theories without definable degree. Our main combinatorial lemma also allows us to repair a mistake in [23].

Indexing (document details)
Advisor: Evans, Lawrence C., Harrington, Leo A.
Commitee: Govindjee, Sanjay, Rezakhanlou, Fraydoun
School: University of California, Berkeley
Department: Mathematics
School Location: United States -- California
Source: DAI-B 71/09, Dissertation Abstracts International
Subjects: Mathematics
Keywords: Amalgamation construction, Comparison principal, Finite difference scheme, Hrushovski's fusion, Infinity Laplacian, Viscosity solution
Publication Number: 3413555
ISBN: 978-1-124-14258-6
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