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Graduate Seminar

Son Van
Carnegie Mellon University
Title: Laplacians and games

Abstract: One of the most studied problems in analysis is to minimize the functional $$I_2[q]= \bigg(\int_{\Omega} \vert Dq\vert^2\bigg)^{1/2}$$ where $\Omega\subseteq \mathbb{R}^n$ is nice and bounded and $q$ is in some admissible set of functions. At least, according to Wikipedia, the minimizer of $I_2$ must satisfy the so-called Euler-Lagrange equation, which, in this case, is the Laplace's equation $$-\Delta q = 0.$$ Now, imagine, what happens if we replace $2$ by $p$ and then, after some normalization, send $p\to\infty$? At least, at the intuitive level, we would be expecting to minimize $$I_{\infty}[q] = \Vert Dq\Vert_\infty,$$ which is the same as finding a function that has the least Lipschitz constant, given some conditions. Based on the same heuristic argument, one can see that the minimizer of $I_\infty$ must satisfy the nonlinear partial differential equation $$\Delta_\infty q:= \frac{1}{{\vert Dq\vert^2}}\sum_{i,j=1}^n q_{x_i}q_{x_j}q_{x_ix_j} =0.$$ $\Delta_\infty$ is called the $\infty$-Laplacian. In this talk, we will try to give a brief overview about this operator and its surprising connection to game theory.

Date: Tuesday, February 14, 2017
Time: 5:30 pm
Location: Wean Hall 8220
Submitted by:  Yangxi Ou
Note: No video recorded.