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On the infinity Laplacian and Hrushovski's fusion


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 Max-Ball 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 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 $f \equiv 0$, $\inf f > 0$, 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 \cite{MR2137000}. 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 \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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