Jacobian determinants for nonlinear gradient of planar $\infty$-harmonic functions and applications

• 主讲人：Yuan Zhou（Beijing normal university）
• 举办方： Beijing-Saint Petersburg Mathematics Colloquium
• 时间： 2024-04-11 20:00 - 2024-04-11 21:00
• 地点： online

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Abstract: In dimension 2, we introduce a distributional Jacobian determinant   for the nonlinear complex gradient $V_\beta(Dv)$ of a function $v\in  W^{1,2 }_\loc$ with $\beta |Dv|^{1+\beta}\in W^{1,2}_\loc$, where $\beta>-1$.  This is new when $\beta\ne0$.  Given any planar $\infty$-harmonic function $u$, we show that such distributional Jacobian determinant $\det DV_\beta(Du)$ is a nonnegative Radon measure with some quantitative local lower and upper bounds. Denoting by $u_p$ the $p$-harmonic function having the same nonconstant boundary condition as $u$, we show that $\det DV_\beta(Du_p) \to \det DV_\beta(Du)$ as $p\to\infty$ in the weak-$\star$ sense in the space of Radon measure. Recall that $V_\beta(Du_p)$ is always quasiregular mappings, but $V_\beta(Du)$ is not in general.

Bio: Yuan Zhou is a professor from Beijing normal university. He is interested in  function spaces, quasiconformal mappings, nonsmooth analysis in metric measure space, quasilinear equations, semilinear equations and also  infinity  Laplacian and Aronsson equations. In particular, he contributed to the second order regularity of planar infinity harmonic functions,  the coincidence of distance and differential structures of metric spaces, and quasiconformality of Triebel-Lizorkin spaces.