 Main
On the infinity Laplacian and Hrushovski's fusion
 Smart, Charles Krug
 Advisor(s): Evans, Lawrence C;
 Harrington, Leo A
Abstract
We study viscosity solutions of the partial differential equation $$ \Delta_\infty u = f \quad \mbox{in } U,$$ where $U \subseteq \R^n$ is bounded and open, $f \in C(U) \cap L^\infty(U)$, and $$\Delta_\infty u := Du^{2} \sum_{ij} u_{x_i} u_{x_i} u_{x_i x_j}$$ is the {\em infinity Laplacian}.
Our first result is the MaxBall Theorem, which states that if $u \in USC(U)$ is a viscosity subsolution of $$ \Delta_\infty u \leq f \quad \mbox{in } U$$ and $\ep > 0$, then the function $v(x) := \max_{\bar B(x,\ep)} u$ satisfies $$2 v(x)  \max_{\bar B(x,\ep)} v  \min_{\bar B(x,\ep)} v \leq \max_{\bar B(x,2 \ep)} f,$$ for all $x \in U_{2\ep} := \{ x \in U : \dist(x, \partial U) > 2\ep \}$. The lefthand side of this latter inequality is a monotone finite difference scheme that is comparatively easy to analyze. The MaxBall 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 $f \equiv 0$, $\inf f > 0$, or $\sup f < 0$. The results mentioned so far are joint work with S. Armstrong.
The MaxBall 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 \cite{MR2137000}. We also present a new adaptive finite difference scheme.
We also prove some results in Model Theory. We study rankpreserving interpretations of theories of finite Morley rank in strongly minimal sets. In particular, we partially answer a question posed by Hasson \cite{MR2286641}, showing that definable degree is not necessary for such interpretations. We generalize Ziegler's fusion of structures of finite Morley rank \cite{MR2441382} to a class of theories without definable degree. Our main combinatorial lemma also allows us to repair a mistake in \cite{MR2354912}.
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