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We study viscosity solutions of the partial differential equation [special characters omitted] where U ⊆ [special characters omitted] is bounded and open, f ∈ C( U) ∩ L∞ (U), and [special characters omitted] is the infinity Laplacian.
Our first result is the Max-Ball Theorem, which states that if u ∈ USC(U) is a viscosity subsolution of [special characters omitted] and ϵ > 0, then the function v( x) := maxB¯( x,ϵ) u satisfies [special characters omitted] for all x ∈ U 2ϵ := {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].
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 |
Source Type: | DISSERTATION |
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 |