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| 開催情報 | 木曜日 16:00～17:30 数理科学研究科棟(駒場) 002号室 |
|---|---|
| 担当者 | 石毛 和弘 |
2024年03月21日(木)
16:00-17:30 数理科学研究科棟(駒場) 126号室
Mostafa Fazly 氏 (University of Texas at San Antonio)
Symmetry Results for Nonlinear PDEs (English)
Mostafa Fazly 氏 (University of Texas at San Antonio)
Symmetry Results for Nonlinear PDEs (English)
[ 講演概要 ]
The study of qualitative behavior of solutions of Partial Differential Equations (PDEs) started roughly in mid-18th century. Since then scientists and mathematicians from different fields have put in a great effort to expand the theory of nonlinear PDEs. PDEs can be divided into two kinds: (a) the linear ones, which are relatively easy to analyze and can often be solved completely, and (b) the nonlinear ones, which are much harder to analyze and can almost never be solved completely.
We begin this talk by an introduction on foundational ideas behind the De Giorgi’s conjecture (1978) for the Allen-Cahn equation that is inspired by the Bernstein’s problem (1910). This conjecture brings together three groups of mathematicians: (a) a group specializing in nonlinear partial differential equations, (b) a group in differential geometry, and more specially on minimal surfaces and constant mean curvature surfaces, and (c) a group in mathematical physics on phase transitions. We then present natural generalizations and counterparts of the problem. These generalizations lead us to introduce certain novel concepts, and we illustrate why these novel concepts seem to be the right concepts in the context and how they can be used to study particular systems and models arising in Sciences. We give a survey of recent results.
The study of qualitative behavior of solutions of Partial Differential Equations (PDEs) started roughly in mid-18th century. Since then scientists and mathematicians from different fields have put in a great effort to expand the theory of nonlinear PDEs. PDEs can be divided into two kinds: (a) the linear ones, which are relatively easy to analyze and can often be solved completely, and (b) the nonlinear ones, which are much harder to analyze and can almost never be solved completely.
We begin this talk by an introduction on foundational ideas behind the De Giorgi’s conjecture (1978) for the Allen-Cahn equation that is inspired by the Bernstein’s problem (1910). This conjecture brings together three groups of mathematicians: (a) a group specializing in nonlinear partial differential equations, (b) a group in differential geometry, and more specially on minimal surfaces and constant mean curvature surfaces, and (c) a group in mathematical physics on phase transitions. We then present natural generalizations and counterparts of the problem. These generalizations lead us to introduce certain novel concepts, and we illustrate why these novel concepts seem to be the right concepts in the context and how they can be used to study particular systems and models arising in Sciences. We give a survey of recent results.
