Research Interests

My research lies in nonlinear analysis  and stochastic processes. I am particularly interested in what happens when a limiting process retains information that a classical description cannot capture: a diffusion acquires mass on the boundary, a thin layer leaves behind an effective boundary law, or a sequence of gradients develops a singular measure. I study how these features enter the equations and energies of a model, and which structural conditions allow them to persist or force them to disappear.

My current work develops these questions in two main directions.

   A boundary can carry its own dynamics. Sticky Brownian motion provides a concrete example: the process spends positive amounts of time on the boundary, and its evolution involves the exchange of probability between the interior and the boundary. Understanding this exchange leads to coupled bulk–boundary equations and dynamic boundary conditions.

  I study these processes through probabilistic constructions, infinitesimal generators, transition kernels, spectral representations, and numerical methods. A central question is how interactions confined to a thin region near the boundary survive in a continuum limit. Which features of the microscopic interaction determine the resulting boundary dynamics, and how can these dynamics be described consistently from probabilistic and PDE perspectives?   In ongoing joint work with Jean-Luc Thiffeault, I investigate sticky Brownian motion and effective boundary dynamics, as well as generalized Navier-like boundary conditions for nonlinear flow systems. These projects motivate my broader interest in how boundary interactions influence transport and dissipation.

  In variational problems with linear coercivity, bounded energy may coexist with gradients concentrating on increasingly small sets. The natural limiting object is then a function of bounded variation, whose derivative can contain both an absolutely continuous part and a singular part.

   I study relaxation and regularity for quasiconvex functionals with (1,q)-growth, including joint work with Franz Gmeineder. My questions concern how the relaxed energy accounts for concentration, how its minimizers satisfy variational equations, and which assumptions recover Sobolev regularity.  A related direction concerns the rigidity of singular derivatives. Here I investigate when differential constraints and nonlinear observations are incompatible with a nonzero singular part. This brings the geometry of singular derivatives into direct contact with regularity: the aim is to identify the mechanisms that permit singular behavior and those that exclude it.

My earlier work on reaction–diffusion equations, free-boundary problems, fractional operators, and nonlocal dispersal models was motivated largely by biological and epidemiological applications. This work provided a broad foundation in qualitative PDE analysis and led me toward more structural questions concerning boundary dynamics, singularities, and limiting processes.

Work in Progress

1. Generalized Navier-like boundary conditions for nonlinear flow systems (with Jean-Luc Thiffeault).

2. Sticky Brownian Motion and effective boundary dynamics (with Jean-Luc Thiffeault).

3. Lower-order regularity for relaxed minimizers under (1, q)-growth (with Franz Gmeineder), 50 pages.

4. Logarithmic slicing and the sharp ellipticity threshold for BV minimizers (with Franz Gmeineder), 27 pages.

Publications/Preprints

1. Pathwise transversality and rigidity of singular higher-order derivatives, 65 pages,  preprint, submitted.

2. Quasiconvex functionals with (1,q)-growth in BV, 37 pages, preprint, submitted.

3. Existence and uniqueness of positive solution for a fractional p-Laplacian equation of logistic-type with harvesting terms (with Khieu T. Tran), Mediterranean Journal of Mathematics, Issue 3, Volume 23, Article No.: 99 (2026), DOI.

4. Stability estimates for nonlinear backward diffusion equations with superposition operators of mixed orders and time-dependent coefficient (with Khieu T. Tran and Khanh Q. Tra), Inverse Problems, Issue 3, Volume 42 (2026), DOI.

5.  On a bistable delayed nonlinear reaction-diffusion equation with a two-phase free boundary: Semi-wave and its numerical simulation, Studies in Applied Mathematics, Issue 2, Volume 154 (2025), DOI.

6.  Monotonicity for quasilinear elliptic problems with a sign-changing nonlinearity in half-planes (with Phuong Le and Chi T. Vo), Comptes Rendus Mathématique. Académie des Sciences. Paris, Volume 363 (2025), pp. 69-87, DOI.

7.  Dynamics for advective-cooperative system with free boundaries in a nondegenerate epidemiological model, Nonlinear Dynamics, Issue 3, Volume 113 (2024), pp. 2941–2968, DOI.

8.  On the fractional p-q Laplace operator with weights  (with Khieu T. Tran), Applicable Analysis, Issue 7, Volume 104 (2023),  pp.1314-1335, DOI.

9.   Dynamics for a two-phase free boundary system in an epidemiological model with couple nonlocal dispersals (with Hung  H. Vo), Journal of Differential Equations, Volume 335 (2022), pp.398–463, DOI.

10.  Spreading of two competing species in advective environment governed by free boundaries with a given moving boundary (with Trong D. Dang and  Hung  H. Vo), Vietnam Journal of Mathematics, Issue 4, Volume 49 (2021), pp.1199–1225, DOI.

Other/unpublished notes

1. Principal eigenvalue and positive solutions for fractional p-q Laplace operator in quantum field theory (2020), arXiv:2006.03233.

2. The impact of fast diffusion on KPP wave propagation in a half-plane,  B.S. Thesis, University of Science, Ho Chi Minh City, Vietnam.