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PDE実解析研究会
過去の記録 ～10/03｜次回の予定｜今後の予定 10/04～
| 開催情報 | 火曜日 10:30～11:30 数理科学研究科棟(駒場) 056号室 |
|---|---|
| 担当者 | 儀我美一、石毛和弘、三竹大寿、米田剛 |
| セミナーURL | http://coe.math.sci.hokudai.ac.jp/sympo/pde_ra/ |
| 目的 | 首都圏の偏微分方程式、実解析の研究をさらに活発にするために本研究会を東大で開催いたします。 偏微分方程式研究者と実解析研究者の討論がより日常的になることを目指しています。 そのため、講演がその分野の概観をもわかるような形になるよう配慮いたします。 また講演者との1対1の討論がしやすいように講演は火曜の午前とし、午後に討論用の場所を用意いたします。 この研究会を通して皆様に気楽に東大を訪問していただければ幸いです。 北海道大学のHPには、第1回(2004年9月29日)～第38回(2008年10月15日)の情報が掲載されております。 |
2019年06月04日(火)
10:30-11:30 数理科学研究科棟(駒場) 056号室
Giuseppe Mingione 氏 (Università di Parma)
Recent progresses in nonlinear potential theory (English)
Giuseppe Mingione 氏 (Università di Parma)
Recent progresses in nonlinear potential theory (English)
[ 講演概要 ]
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.
