Description of some of my research directions

This is a preliminary version (2/2025). It will be updated more as time goes.

1. Properties of viscosity solutions of Hamilton-Jacobi equations and their connections to optimal control and dynamical systems

A great resource to this is the list of open questions composed by my colleague, Sergey Bolotin [list]. See also Section 5 in the short lecture notes by Craig Evans [Ev]. It is known now that the periodic homogenization theory of Hamilton-Jacobi equations and weak KAM (Aubry-Mather) theory are much related through the cell (ergodic) problems, studied in an unpublished paper of Lions, Papanicolaou, and Varadhan [LPV].

1.1. Homogenization

In Paper 31, we study further the aforementioned connection and obtain the optimal rate of convergence in various situations in periodic homogenization of Hamilton-Jacobi equations in convex setting. We find a natural connection between how fast the average of backward characteristic converging to its rotation vector and rates of convergence in the homogenization problem. In Paper 40, we continue exploring this connection for the homogenization of first-order front propagation problems. We show that if the effective front is a polytope, then we have optimal rate of convergence.

In Paper 52, we finally obtain the optimal rate of convergence $O(\epsilon)$ for the general convex case. We also get the optimal rate for a class of nonconvex Hamiltonians. We rely on an equal curve cutting lemma. In Paper 68, we develop further this direction and obtain the optimal convergence rate for the state-constraint problems. We then consider a dilute situation in which the holes' diameter is much smaller than the microscopic scale. Finally, a homogenization problem with domain defects where some holes are missing is analyzed.

Optimal rate of convergence in periodic homogenization of viscous Hamilton-Jacobi equations: In Paper 67, we show that the optimal rate in the viscous case is $O(\epsilon^{1/2})$. Further, we study more on the vanishing viscosity process and give various numerical examples.

Stochastic homogenization: See Papers 10, 12, 15, 16, 18, 25, 28, 37. Papers 12, 18, 28 study stochastic homogenization of nonconvex Hamilton-Jacobi equations (see Question 32 in [list] by Souganidis, which also has been asked by Lions, Varadhan, and some others). Besides, homogenization in dynamic random environment is quite tricky, and there are many questions to be studied.

Periodic homogenization: See Papers 6, 31, 40, 52, 53, 56, 61, 67, 68.

1.2. Nonconvex Hamilton-Jacobi equations

I am extremely interested in nonconvex Hamilton-Jacobi equations (see Papers 1, 2, 12, 18, 24, 28, 39). Paper 2 addresses weak KAM theory in the nonconvex setting (see nonconvex Hamiltonians in Section 5 of [Ev], and a related question, Question 11 in [list] by Albert Fathi).

Paper 28 is the first attempt to systematically study properties of the effective Hamiltonian arising in the periodic homogenization of some coercive but nonconvex Hamilton-Jacobi equations. Firstly, we introduce a new and robust decomposition method to obtain min-max formulas for a class of nonconvex effective Hamiltonians. Secondly, we analytically and numerically investigate other related interesting phenomena, such as "quasi-convexification" and breakdown of symmetry. Finally, in the appendix, we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for same class of nonconvex Hamilton-Jacobi equations. Some conjectures and problems are also proposed. We are just at the beginning of this direction. Papers 12, 18 belong to this direction.

In Paper 52, we get the optimal rate of convergence in periodic homogenization for a class of nonconvex Hamiltonians. See a conjecture (Summer 2022) I posed in this direction pdf.

1.3. Cell (ergodic) problems

Up to now, the properties of the cell problems and the effective Hamiltonians are still quite mysterious and not yet well-studied. As it was shown explicitly in an example in [LPV], the cell problem does not have unique solutions (even up to additive constants) and thus how can we select one solution out of these many? This leads naturally to a question on the selection problem of the vanishing discount approximation procedure. Roughly speaking, does the vanishing discount procedure select one and only one solution out of these infinitely many solutions of the cell problem? In Paper 17, we get the convergence result for degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians. In Papers 23 and 26, we have been able to handle the general convex, possibly degenerate fully nonlinear elliptic equations under various types of boundary conditions (periodic, state constraint, Dirichlet, Neumann boundary conditions). This means that convex cases are more or less complete. In Paper 24, we have studied some specific nonconvex cases (quasiconvex, double-well type Hamiltonians in one dimension).

In Paper 32, we provide a simple way to find uniqueness sets for ergodic problems of first and second order Hamilton-Jacobi equations by using a PDE approach. Paper 36 is also along this line, in which we deal with a generalized ergodic problem of contact Hamilton-Jacobi equation type.

1.4. New tool: the nonlinear adjoint method

We basically look at the linearized operator the nonlinear equation around the solution and study the adjoint equation to this linearized operator. By using this new adjoint equation and integration by parts, we derived some new estimates and conservation laws. These new estimates are quite helpful in obtaining some fine properties of solutions. See Papers 1, 2, 3, 5, 8, 9, 17, 24, 32, 36, 44.

1.5. Large time behaviors

Large time behavior of solutions to Hamilton-Jacobi equations has received a lot attention recently. See Papers 4, 7, 8, 9. In Papers 8, 9, we were able to obtain the large time behavior result for viscous Hamilton-Jacobi equations with possibly degenerate diffusions and uniformly convex Hamiltonians. In Paper 44, we obtained the large time profile for Hamilton--Jacobi--Bellman equations.

1.6. Policy iterations and optimal control

Policy iteration for the deterministic control problems -- a viscosity approach: Paper 58. To overcome the problem of ill-posedness due to lack of regularity, we consider a semi-discrete scheme by adding a viscosity term via finite differences in space. We prove that PI for the semi-discrete scheme converges exponentially fast, and provide a bound on the error induced by the semi-discrete scheme. We also consider the discrete space-time scheme.

Policy Iteration for exploratory Hamilton--Jacobi--Bellman equations: Paper 69.

Controlling discrete semilinear wave equations toward flocks: Paper 72.

1.7. Other topics

Numerics for Hamilton-Jacobi equations via the convexification method: Paper 47. See also Paper 51.

State-constraint static Hamilton-Jacobi equations in nested domains: In Paper 39, we study quantitative behaviors of state-constraint solutions in the given nested domains. Both convex and nonconvex cases are analyzed.

Viscosity solutions of general viscous Hamilton-Jacobi equations: In Paper 11, we present comparison principles, Lipschitz estimates, and study state-constraint problems for degenerate, second-order Hamilton-Jacobi equations.

Weakly coupled systems and random switchings: See Papers 4, 6, 7, 14, 19. In the convex setting, a weakly coupled system of Hamilton-Jacobi equations corresponds to an optimal control problem, which appears in the dynamic programming, for the system whose states are governed by random changes. The random changes of the states are controlled by a continuous-time Markov chain.

2. Inverse problems and effective quantities

The basic question is that how can we read off microscopic information if we are given macroscopic information? We propose some inverse problems to study properties of the effective Hamiltonians from a different angle (sort of an extrinsic way). A simple way to address this kind of question is that if the two different Hamiltonians give the same effective Hamiltonian, then what can we say about the relations between the two? This has rich connections to dynamical systems, KAM, weak KAM theories. See Papers 20, 30. Paper 20 formulates this kind of questions for the first time. In Paper 30, we prove a rigidity result in two dimensions, which partially resolves a conjecture stated in Paper 20.

In Paper 22, we study the effective burning velocity from a combustion model. The level sets of the effective burning velocity are convex, but not much else is known. We study the non-roundedness, flatness of the level sets of the effective burning velocity, which is a type of inverse problems. A similar question to that of Paper 20 is also studied. Paper 28 also belongs to this category.

In Paper 40, we show that in three dimensions or higher, one can always obtain effective fronts of centrally symmetric polytopes with rational coordinates and nonempty interiors.

In Paper 53, we show that in two dimensions, one can always obtain effective fronts of centrally symmetric polygons with rational coordinates and nonempty interiors. The two dimensional case is much harder because of the topological restriction.

In Paper 56, we study the differentiability of effective fronts in the continuous setting in two dimensions. Our main result says that that the boundary of the effective front is differentiable at every irrational point. Combining with the sufficiency result in Paper 53, our result implies that for continuous coefficients, a polygon could be an effective front if and only if it is centrally symmetric with rational vertices and nonempty interior.

3. Level-set Mean Curvature Flows

Level-set mean curvature flows and crystal growth: See Papers 21, 34, 38, 65. We are interested in the asymptotic growth speed of crystals. The equation of interest is of birth and spread type nonlinear PDEs. In Papers 21, 34, we show that the asymptotic growth speed exists under fairly general assumptions. The asymptotic growth speed is the first term in the asymptotic expansion of the solution. Then, in Paper 38, we study large time behavior of the solution in some specific cases. This is roughly the next term in the asymptotic expansion of the solution, and the large time behavior is quite complicated since the equation is degenerate and not convex in the gradient variable. The general case is still open.

For the spiral crystal growth model, we provide a systematic study in Paper 65. We obtain the uniform Lipschitz estimate of the solution (under a strong condition on the forcing term $c$), the existence of the asymptotic growth rate $S_c$, some estimates of $S_c$ and its strong dependence on the boundary of $W$. We also understand that this problem is very different from the birth-and-spread model because of the failure of large time behaviors through a clear example.

Related problems: See Paper 35 (obstacle problem for mean curvature flow with driving force).

In Paper 50, we study the level-set forced mean curvature flow with the Neumann boundary condition. We first show that the solution is Lipschitz in time and locally Lipschitz in space. Then, under an additional condition on the forcing term, we prove that the solution is globally Lipschitz. We obtain the large time behavior of the solution in this setting and study the large time profile in some specific situations. Finally, we give two examples demonstrating that the additional condition on the forcing term is sharp, and without it, the solution might not be globally Lipschitz.

In Paper 61, we study the bifurcation of homogenization and nonhomogenization of the curvature G-equation with shear flows. We show that the effective burning velocity associated with shear flows in dimensions three or higher ceases to exist when the flow intensity surpasses a bifurcation point. The characterization of the bifurcation point in three dimensions is closely related to the regularity theory of two-dimensional minimal surface type equations. As a consequence, a bifurcation also exists for the validity of full homogenization of the curvature G-equation associated with shear flows.

Quenching for axisymmetric hypersurfaces under forced mean curvature flows: Paper 66.

4. Coagulation-Fragmentation equations

We are interested in a critical Coagulation-Fragmentation (C-F) equation and study the wellposedness of mass-conserving solutions. More precisely, we focus on the case of multiplicative coagulation kernel and constant fragmentation kernel. For this particular case, there is a sharp conjecture saying that the result depends on the mass $m(0)$ of the initial data. In particular, the conjecture says that if $m(0)>1$, then there is no mass-conserving solutions; and if $m(0) \leq 1$, then there is a unique mass-conserving solution.

In Paper 41, we use the Bernstein transform to transform the critical C-F equation to a singular Hamilton-Jacobi equation. This critical Hamilton-Jacobi equation is new in the literature of viscosity solutions. We prove that indeed, if $m(0)>1$, then there is no mass-conserving solutions. We show that if $m(0) \leq 1$, then there is at most one mass-conserving solution. Then, we obtain existence of such a mass-conserving solution for $0< m(0) < 1/2$. The existence of such solutions in case where $1/2 \leq m(0) \leq 1$ is still open.

In Paper 43, we study large time behavior for this critical Hamilton-Jacobi equation in the critical mass case $m(0)=1$. Thus, large time behavior is well understood for all $0< m(0) \leq 1$.

In Paper 55, we study further the case $m(0)>1$ and prove that there exists a unique mass-conserving solution locally in time. Clearly, in this case, there is no global mass-conserving solution and hence the understanding is rather complete in this case.

In Paper 71, we study the discrete critical case. The discrete case turns out to be trickier. We apply the discrete Bernstein transform to the original Coagulation-Fragmentation equation to get two new singular Hamilton-Jacobi equations and use viscosity solution methods to analyze them. We need to use both of the new equations to perform our analysis and show that if $m(0)>1$, then there is no mass-conserving solutions. If $m(0) \leq 1$, then there is at most one mass-conserving solution. Then, we obtain existence of such a mass-conserving solution for $0< m(0) < 1/2$. What we can do further here is to obtain large time behavior of the solution for the CF equation when $0< m(0) < 1/2$. (This is stronger than the continuous cases).

5. Homogenization of non-divergence form linear elliptic PDE

5.1. Periodic case

In Papers 42, 46, we study optimal rate of convergence in periodic homogenization theory of non-divergence form linear elliptic PDE. Among the results, we show that the optimal rate in $L^\infty$ norm is $O(\epsilon)$ generically. More precisely, the set of diffusion matrices that give $O(\epsilon)$-rate (so-called $\epsilon$-type) is open and dense. We also give rate of convergence in $W^{1,p}$ and $W^{2,p}$ norms and numerical illustrations for the optimality of these convergence rates.

A conjecture (Summer 2020) related to Papers 42, 46, pdf.

This conjecture was proved in Paper 54. Further analysis of the diffusion matrices ($\epsilon^2$-type) were studied here.

5.2. Balanced random environments

Quantitative homogenization: Paper 37.

Stochastic integrability of heat kernel bounds: Paper 57. We first obtain an exponential integrability of the heat kernel bounds. We then prove the optimal diffusive decay of the semigroup generated by the heat kernel for $d\geq 3$.

Optimal convergence rates in stochastic homogenization in a balanced random environment: Paper 59. We show that the optimal rate is $O(\epsilon)$ when $d \geq 3$. And when $d=2$, we obtain a nearly optimal rate.

6. Dynamical systems and FLRW-cosmology

Late-time cosmological solutions and bound of the $\epsilon$-parameter: Papers 60, 62.

Analytic bounds on the equation of state of a cosmological fluid on general effective theories of ekpyrosis: Paper 64.

Analytic bounds on late-time axion-scalar cosmologies: Paper 70.

7. Misc

7.1. Applications to Large Deviations in Games

See Papers 33, 45. We characterize solutions of a class of time-homogeneous optimal control problems with semilinear running costs and state constraints as maximal viscosity subsolutions to Hamilton-Jacobi equations, and show that optimal solutions to these problems can be constructed explicitly in some cases. We present applications to large deviations problems arising in evolutionary game theory.

7.2. Langevin equation with variable friction

See Paper 29. We study two asymptotic problems for the Langevin equation with variable friction coefficient. The first is the small mass asymptotic behavior, known as the Smoluchowski-Kramers approximation, of the Langevin equation with strictly positive variable friction. The second result is about the limiting behavior of the solution when the friction vanishes in regions of the domain. Previous works on this subject considered one dimensional settings with the conclusions based on explicit computations.

7.3. Mean Field Games

In Paper 27, I introduce a nonconvex Mean Field Games system by studying a model with a large number of identical pairs of players who are all rational, and each pair plays an identical zero-sum differential game. Wellposedness is then studied for a simple system of this kind.

7.4. Calculus of variations and regularity

Partial regularity for minimizers of singular energy functionals and application to liquid crystal: See Paper 13.

7.5. Degenerate linear parabolic equations

Degenerate linear parabolic equations in divergence form: Papers 48, 49. Degenerate linear parabolic equations in nondivergence form: Paper 63. We obtain the wellposedness and regularity of solutions in weighted Sobolev spaces ($W^{1,p}$, $W^{2,p}$ weighted estimates and related results).