## PDE Real Analysis Seminar

Seminar information archive ～02/25｜Next seminar｜Future seminars 02/26～

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.)

Recent progresses in nonlinear potential theory (English)

**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.

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.