Research Interests

A particular focus is on sticky Brownian motion and related diffusion processes that spend positive amounts of time on the boundary. Their probability laws naturally lead to coupled bulk–boundary evolution equations and dynamic boundary conditions. I study these models through probabilistic constructions, infinitesimal generators, bulk–boundary transition kernels, spectral representations, and numerical approximations. A broader objective is to understand how microscopic or thin-layer boundary dynamics generate effective continuum boundary laws, including Navier-type boundary models in fluid mechanics.

Here, I focus on relaxation and regularity for quasiconvex variational integrals with nonstandard growth, particularly functionals with (1,q)-growth in BV. In this borderline regime of linear coercivity, the nonreflexivity of  W1,1, together with the possible formation of concentrations and singular components of Du, prevents a direct application of standard Sobolev compactness and regularity methods. This calls for relaxation in BV and refined variational techniques adapted to measure-valued derivatives. My work investigates the resulting relaxed energies and generalized minimizers, their Euler–Lagrange structure, and their qualitative and regularity properties, with motivations from nonlinear elasticity, continuum mechanics, and materials science.

My earlier work concerned nonlinear PDEs arising from free-boundary problems, reaction–diffusion systems, fractional operators, and nonlocal dispersal models. I studied existence, uniqueness, monotonicity, spreading dynamics, long-time behavior, and the effects of diffusion, delay, advection, and nonlocal interactions. This work provided a broad foundation in nonlinear analysis and continues to inform my current research.

Work in Progress

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

2. Dimension and structure of singularities for strongly quasiconvex functionals of (1,q)-growth,  In preparation.

Submitted Manuscripts

3.  Lower-order regularity for relaxed minimizers under (1, q)-growth, Submitted, 2026. 52 pp.

2.  Nonlinear transversality for BV fields and rigidity of singular Hessians, Submitted, 2026. 10 pp.

1.  Quasiconvex functionals with (1, q)-growth on BV, Submitted, 2026. 37 pp.

Publications

8. 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.

7. 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.

6.  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.

5.  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.

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

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

2.   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.

1.  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

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

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