## Past Talks

Motion by mean curvature and coupled KPZ from particle systems
Monday, 25 May 2020.
Professor @Waseda University
Funaki was a Professor @The University of Tokyo from 1995 until 2017 and is now a Professor @Waseda University from 2017. He is a Professor Emeritus of the University of Tokyo.

#### Abstract

It's one of common interests in probability group to derive nonlinear PDEs or stochastic PDEs from microscopic interacting systems via scaling limits in space and time, usually by some averaging effect caused by the local ergodic property of the microscopic systems.

In the talk, we discuss the derivation of two objects: Motion by mean curvature (MMC) and coupled KPZ (Kardar-Parisi-Zhang) equation. The microscopic system we take is that of particles which perform random walks with interaction. To derive MMC, we allow creation and annihilation of particles. The system exhibits a phase separation to sparse and dense regions of particles and, macroscopically, the interface separating these two regions evolves under the MMC. We pass through the Allen-Cahn equation with nonlinear Laplacian at an intermediate level. On the other hand, the coupled KPZ equation is a system of nonlinear singular stochastic PDEs. It is ill-posed in a classical sense and requires renormalizations. We derive it under the fluctuation limit of multi-species weakly-asymmetric particle system of interacting random walks without creation and annihilation. The so-called Boltzmann-Gibbs principle plays a fundamental role for both.

References

The first part is joint work with S. Sethuraman, D. Hilhorst, P. El Kettani and H. Park ( arXiv:2004.05276), while the second is with C. Bernardin and S. Sethuraman ( arXiv:1908.07863).

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The vanishing discount problem for systems of Hamilton-Jacobi equations
Monday, 18 May 2020.
Research Fellow @Tsuda University (Japan), Professor Emeritus @Waseda University (Japan), Fellow @American Mathematical Society
Professor Ishii received his Ph.D. from Waseda University in June 1975. He worked at Chuo University from 1976 until 1996, at Metropolitan University from 1996 until 2001, and at Waseda University from 2001 until 2018.

#### Abstract

In the talk, I discuss the recent developments of the vanishing discount problem. The main focus concerns that for monotone systems of Hamilton-Jacobi equations. The principal tool in the analysis for the vanishing discount is the use of Mather measures from Aubry-Mather theory or their generalizations. I explain a way of constructing Mather measures and an approach to the vanishing discount problem.

This talk bases on joint work with Liang Jin of Nanjing University of Science and Technology.

References
• [1]  Hitoshi Ishii and Liang Jin, The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 2: nonlinear coupling, arXiv:1906.07979 [math.AP], to appear in Calculus of Variations and PDE's.
• [2]  Hitoshi Ishii, The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling, arXiv:1903.00244 [math.AP].
• [3]  Andrea Davini and Maxime Zavidovique, Convergence of the solutions of discounted Hamilton-Jacobi systems, Adv. Calc. Var. Online publication (2019), DOI: 10.1515/acv-2018-0037.

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Recent progress on a sharp lower bound for Steklov eigenvalue
Monday, 11 May 2020.
Professor
@ Xiamen University, China
Professor Xia received his PhD in July 2012 from the Albert Ludwig University of Freiburg in Germany. He was a doctoral student of Professor Guofang Wang. Professor Xia was a Postdoc Fellow at the Max Planck Institute for Mathematics and Sciences at Leipzig in Germany from 2012-2014, and Postdoc Fellow at McGill University in Montreal, Canada, from 2015-2016.

#### Abstract

Lichnerowicz-Obata's theorem says that for a closed $n$-manifold with Ricci curvature $Ric\ge (n-1)K>0$, the first (nonzero) eigenvalue is greater than or equal to $nK$, with equality holding only on a round $n$-sphere. Similar results for the first Dirichlet eigenvalue and Neumann eigenvalue have been shown by Reilly, Escobar and C.Y.Xia respectively. For the first (nonzero) Steklov eigenvalue, a conjecture has been made by Escobar saying that for a compact manifold with boundary which has nonnegative Ricci curvature and boundary principal curvatures bounded below by some $c>0$, the first (nonzero) Steklov eigenvalue is greater than or equal to $c$, with equality holding only on a Euclidean ball. This conjecture is true in two dimensions due to Payne and Escobar. In this talk, we present a resolution to this conjecture in the case of nonnegative sectional curvature in any dimensions. We will also discuss a sharp comparison result between the first (nonzero) Steklov eigenvalue and the boundary first eigenvalue.

This is a joint work with Changwei Xiong.