Main Research Line

My research lies at the intersection of nonlinear differential equations, variational methods, functional analysis, and geometric techniques. I study ordinary and partial differential equations, with particular emphasis on the existence, multiplicity, stability, symmetry, rigidity, and asymptotic behavior of their solutions.

Transition Solutions and Rigidity for Allen–Cahn-Type Equations

My main research project concerns transition solutions for nonlinear equations and systems of the general form


-L(u)+a(x)W'(u)=0,

where W is a double-well or multi-well potential, a(x) describes the spatial heterogeneity, and L may be a linear, quasilinear, nonlocal, or graph-based operator. The project investigates autonomous and non-autonomous problems and seeks general conditions ensuring the existence, multiplicity, classification, and qualitative behavior of heteroclinic, homoclinic, periodic, layered, saddle-type, brake, and multi-bump solutions. Particular attention is given to variational constructions, constrained minimization, first-integral methods, compactness mechanisms, and the influence of the geometry of the potential and of the coefficient a(x). Another central direction concerns rigidity and symmetry phenomena for Allen–Cahn-type equations. This includes Modica-type gradient estimates, Liouville properties, one-dimensional symmery, and questions related to the De Giorgi and Gibbons conjectures. The aim is to understand how analytical assumptions, asymptotic conditions, and the geometry of level sets force multidimensional solutions to reduce to one-dimensional transition profiles. The project also considers extensions to quasilinear and nonlocal operators, metric and discrete graphs, and geometrically structured spaces.

Other Research Interests

Nonlinear Schrödinger Equations and Normalized Solutions

I study nonlinear Schrödinger-type equations under prescribed mass constraints, with emphasis on variational methods, energy minimization, mass thresholds, concentration phenomena, stability, and loss of global minimality. My interests include equations on Euclidean spaces and metric graphs, general nonlinearities, and models involving nonlocal, Choquard-type, and other generalized operators.

Nonlinear Equations on Metric and Discrete Graphs

I investigate how the topology and geometry of graphs influence the existence and qualitative properties of solutions to nonlinear differential equations. Current topics include normalized solutions on compact and periodic metric graphs, transition solutions on network structures, spectral problems for nonlinear graph operators, and variational and spectral aspects of the \Phi-Laplacian on finite weighted graphs.

Dissipative Gradient Systems and Evolution Equations

I am interested in second-order gradient systems with damping and in related evolutionary PDEs. My research addresses Lyapunov methods, uniform and asymptotic stability, convergence to equilibria, instability near saddle points, and the existence of connecting trajectories. I am also interested in extensions to damped wave equations and systems with time-dependent dissipation.

Geometric Analysis and Rigidity

My work in geometric analysis focuses on the interaction between elliptic equations and the geometry of the underlying space. Topics of interest include overdetermined problems, drift Laplacians, curvature effects, geometric properties of level sets, symmetry results, and rigidity phenomena for nonlinear elliptic equations.

Secondary Research Topics

Number Theory and Density of Sets of Integers

I investigate natural and asymptotic densities of subsets of the positive integers, including relative densities and generalized notions of density determined by comparison scales. My current interests include counting functions, the classification and comparison of density-zero sets, parametrizations of integer subsets, and connections with additive and combinatorial number theory.

History, Exposition, and Education in Mathematics

I also develop expository materials, surveys, and minicourses designed to present advanced mathematics in a conceptually clear and historically informed manner. In mathematics education, my interests include the preparation of future teachers and the creation of resources that: