# Research

I am interested in Partial Differential Equations. In particular, I am working on problems related to Hamilton-Jacobi equation.

## Preprints and Publications

1. Vanishing discount problems for Hamilton--Jacobi equations on changing domains, (submitted)
2. We study the asymptotic behavior, as $\lambda\rightarrow 0^+$, of the state-constraint Hamilton--Jacobi equation $$\begin{cases} \phi(\lambda) u_\lambda(x) + H(x,Du_\lambda(x)) \leq 0 \qquad\text{in}\,\;(1- r(\lambda))\Omega,\\ \phi(\lambda) u_\lambda(x) + H(x,Du_\lambda(x)) \geq 0 \qquad\text{on}\;(1- r(\lambda))\overline{\Omega}. \end{cases} \tag{S_\lambda}$$ Here, $\Omega$ is a bounded domain of $\mathbb{R}^n$ and $\phi(\lambda),r(\lambda):(0,\infty)\rightarrow (0,\infty)$ are continuous nondecreasing functions such that $\lim_{\lambda\rightarrow 0^+} \phi(\lambda) = \lim_{\lambda\rightarrow 0^+} r(\lambda) = 0$. A similar problem on $(1+r(\lambda))\Omega$ is also considered. Surprisingly, we are able to obtain both convergence results and non-convergence results in this setting. Moreover, we provide a very first result on the asymptotic expansion of the additive eigenvalue of $H$ in $(1\pm r(\lambda))\Omega$ as $\lambda\rightarrow 0^+$.
3. (with Yeoneung Kim and Hung V. Tran)
State-constraint static Hamilton-Jacobi equations in nested domains, SIAM journal on Mathematical Analysis
4. We study state-constraint static Hamilton-Jacobi equations in a sequence of domains $\{\Omega_k\}_{k \in \mathbb{N}}$ in $\mathbb{N}^n$ such that $\Omega_k \subset \Omega_{k+1}$ for all $k\in \mathbb{N}$. We obtain rates of convergence of $u_k$, the solution to the state-constraint problem in $\Omega_k$, to $u$, the solution to the corresponding problem in $\Omega = \bigcup_{k \in \mathbb{N}} \Omega_k$. In many cases, the rates obtained are proven to be optimal. Various new examples and discussions are provided at the end of the paper.
5. Rate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimension, Asymptotic Analysis.
6. Let $u^\epsilon$ and $u$ be viscosity solutions of the oscillatory Hamilton-Jacobi equation and its corresponding effective equation. Given bounded, Lipschitz initial data, we present a simple proof to obtain the optimal rate of convergence $\mathcal{O}(\epsilon)$ of $u^\epsilon \rightarrow u$ as $\epsilon \rightarrow 0^+$ for a large class of convex Hamiltonians $H(x,y,p)$ in one dimension. This class includes the Hamiltonians from classical mechanics with separable potential. The proof makes use of optimal control theory and a quantitative version of the ergodic theorem for periodic functions in dimension $n=1$.

## Talks and Presentations

1. Vanishing discount problems for Hamiltonâ€“Jacobi equations on changing domains,
Graduate School of Mathematical Sciences, The University of Tokyo, October 2020.
2. State-constraint static Hamilton-Jacobi equation in nested domains,
Physical Applied Math seminar, University of Wisconsin - Madison, September 2019.
3. State-constraint static Hamilton-Jacobi equation in nested domains,
PDE and Geometric Analysis seminar, University of Wisconsin - Madison, September 2019.
4. Homogenization of first order Hamilton-Jacobi equations,
Specialty Exam, University of Wisconsin - Madison, April 2019.
5. State-constraint static Hamilton-Jacobi equation in nested domains,
Mathematical conference: Summer meeting 2019, University of Science - HCMC, July 2019.
6. Contributed Talk: Some recent works on homogenization of Hamilton-Jacobi equations,
Geometric and Harmonic Analysis 2019, University of Connecticut, March 2019.
7. Homogenization of Hamilton-Jacobi equation: Rate of convergence,
CNA Workshop 2019: Mathematical Models for Pattern fomation, Carnegie Mellon University, March 2019.
8. We study the rate of convergence $u^\varepsilon\longrightarrow u$ as $\varepsilon\longrightarrow 0^+$ in periodic homogenization of Hamilton-Jacobi equations. Here $u^\varepsilon$ and $u$ are viscosity solutions to the oscillatory Hamilton-Jacobi equation \begin{equation*} \begin{cases} u^\varepsilon_t + H\left(x,\frac{x}{\varepsilon},Du^\varepsilon\right) &= 0 \quad\quad\;\,\text{in}\quad \mathbb{R}\times [0,\infty)\\ \qquad\qquad\quad u^\varepsilon(x,0) &= u_0(x) \;\;\text{on}\quad \mathbb{R}, \end{cases}, \qquad \begin{cases} u_t + \overline{H}\left(x,Du\right) &= 0 \quad\quad\;\,\text{in}\quad \mathbb{R}\times [0,\infty)\\ \quad\;\;\,\qquad u(x,0) &= u_0(x) \;\;\text{on}\quad \mathbb{R}, \end{cases} \end{equation*} respectively. We review some results on rate of convergence in the literature and the new result with optimal rate $\mathcal{O}(\varepsilon)$ for a class of convex Hamiltonians in one dimension.
9. Rate of convergence for periodic homogenization of convex Hamilton-Jacobi equations in one dimension,
Physical Applied Math seminar, University of Wisconsin - Madison, September 2018.

## Conferences and Workshops attended

1. High Dimensional Hamilton-Jacobi PDEs Stochastic PDEs, IPAM, UCLA, May 2020
2. High Dimensional Hamilton-Jacobi PDEs Tutorial, IPAM, UCLA, Mar 2020
3. AMS Fall Central Sectional Meeting, University of Wisconsin-Madison, Madison, WI. Sep 2019.
4. Geometric and Harmonic Analysis 2019, University of Connecticut, March 2019.
5. CNA Workshop 2019: Mathematical Models for Pattern fomation, Carnegie Mellon University, March 2019.
6. 81st Midwest PDE seminar, University of Wisconsin-Madison, Madison, WI, April 21-22, 2018.
7. Fourth Chicago Summer School in Analysis, University of Chicago, June 2017.
8. Minischool in Calculus of Variations and Nonlinear Partial Differential Equations, UC - Berkeley, May 2017.
9. Madison workshop in Analysis and PDEs, UW Madison, Oct 1-2, 2016.
10. DAAD Spring School on Combinatorial Stochastic Processes, VIASM, March 2016.
11. Summer Program onPDEs and Applied Mathematics, VIASM, July - August 2014.