Research
I am interested in Partial Differential Equations. In particular, I am working on problems related to HamiltonJacobi equation.Preprints and Publications
 Vanishing discount problems for HamiltonJacobi equations on changing domains, (submitted) Abstract
 (with Yeoneung Kim and Hung V. Tran)
Stateconstraint static HamiltonJacobi equations in nested domains, SIAM journal on Mathematical Analysis
Abstract
 Rate of convergence for periodic homogenization of convex HamiltonJacobi equations in one dimension, Asymptotic Analysis. Abstract
We study the asymptotic behavior, as $\lambda\rightarrow 0^+$, of the stateconstraint HamiltonJacobi equation
\begin{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{$S_\lambda$}
\end{equation}
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 nonconvergence 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 $\lambda\rightarrow 0^+$.
We study stateconstraint static HamiltonJacobi 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 stateconstraint 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.
Let $u^\epsilon$ and $u$ be viscosity solutions of the oscillatory HamiltonJacobi 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

Vanishing discount problems for Hamiltonâ€“Jacobi equations on changing domains,
Graduate School of Mathematical Sciences, The University of Tokyo, October 2020. 
Stateconstraint static HamiltonJacobi equation in nested domains,
Physical Applied Math seminar, University of Wisconsin  Madison, September 2019. 
Stateconstraint static HamiltonJacobi equation in nested domains,
PDE and Geometric Analysis seminar, University of Wisconsin  Madison, September 2019. 
Homogenization of first order HamiltonJacobi equations,
Specialty Exam, University of Wisconsin  Madison, April 2019. 
Stateconstraint static HamiltonJacobi equation in nested domains,
Mathematical conference: Summer meeting 2019, University of Science  HCMC, July 2019.  Contributed Talk: Some recent works on homogenization of HamiltonJacobi equations,
Geometric and Harmonic Analysis 2019, University of Connecticut, March 2019. 
Homogenization of HamiltonJacobi equation: Rate of convergence,
CNA Workshop 2019: Mathematical Models for Pattern fomation, Carnegie Mellon University, March 2019.  Rate of convergence for periodic homogenization of convex HamiltonJacobi equations in one dimension,
Physical Applied Math seminar, University of Wisconsin  Madison, September 2018.
We study the rate of convergence $u^\varepsilon\longrightarrow u$ as $\varepsilon\longrightarrow 0^+$ in periodic homogenization of HamiltonJacobi equations.
Here $u^\varepsilon$ and $u$ are viscosity solutions to the oscillatory HamiltonJacobi 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.
Conferences and Workshops attended
 High Dimensional HamiltonJacobi PDEs Stochastic PDEs, IPAM, UCLA, May 2020
 High Dimensional HamiltonJacobi PDEs Tutorial, IPAM, UCLA, Mar 2020
 AMS Fall Central Sectional Meeting, University of WisconsinMadison, Madison, WI. Sep 2019.
 Geometric and Harmonic Analysis 2019, University of Connecticut, March 2019.
 CNA Workshop 2019: Mathematical Models for Pattern fomation, Carnegie Mellon University, March 2019.
 81st Midwest PDE seminar, University of WisconsinMadison, Madison, WI, April 2122, 2018.
 Fourth Chicago Summer School in Analysis, University of Chicago, June 2017.
 Minischool in Calculus of Variations and Nonlinear Partial Differential Equations, UC  Berkeley, May 2017.
 Madison workshop in Analysis and PDEs, UW Madison, Oct 12, 2016.
 DAAD Spring School on Combinatorial Stochastic Processes, VIASM, March 2016.
 Summer Program onPDEs and Applied Mathematics, VIASM, July  August 2014.