PDE Real Analysis Seminar

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Date, time & place Tuesday 10:30 - 11:30 056Room #056 (Graduate School of Math. Sci. Bldg.)

2019/06/04

10:30-11:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Giuseppe Mingione (Università di Parma)
Recent progresses in nonlinear potential theory (English)
[ Abstract ]
Nonlinear Potential Theory aims at studying the fine properties of solutions to nonlinear, potentially degenerate nonlinear elliptic and parabolic equations in terms of the regularity of the give data. A major model example is here given by the $p$-Laplacean equation
$$ -\operatorname{div}(|Du|^{p-2}Du) = \mu \quad\quad p > 1, $$
where $\mu$ is a Borel measure with finite total mass. When $p = 2$ we find the familiar case of the Poisson equation from which classical Potential Theory stems. Although many basic tools from the classical linear theory are not at hand - most notably: representation formulae via fundamental solutions - many of the classical information can be retrieved for solutions and their pointwise behaviour. In this talk I am going to give a survey of recent results in the field. Especially, I will explain the possibility of getting linear and nonlinear potential estimates for solutions to nonlinear elliptic and parabolic equations which are totally similar to those available in the linear case. I will also draw some parallels with what is nowadays called Nonlinear Calderón-Zygmund theory.