My research focuses on the Analysis & PDEs and Geometric Analysis. Specifically, I am interested in the following:
Functional inequalities: Functional inequalities appear in the study of several PDEs: for instance, they can be shown to be equivalent to smoothing properties of a suitably associated flow. There are many natural questions to ask: from establishing the inequality itself to finding the best constant and proving stability. As a doctoral student, I began my research on this topic, including Hardy, Poincaré, and Rellich-type inequalities, particularly on negatively curved manifolds such as hyperbolic space. This study relies on criticality theory for second-order elliptic operators and the careful application of Bessel pair concepts. Later, I developed an interest in examining these inequalities in Euclidean space as well as in discrete settings. Recently, I also became interested in studying these inequalities in the context of sub-elliptic operators generated by Grushin vector fields. These operators serve as a crucial link between elliptic and non-elliptic theory and represent a fundamental example in sub-Riemannian geometry.
Differential equations: My research focuses on Hamilton-Jacobi equations that arise in specific ergodic control problems involving diffusions in a switching environment. I have studied the existence and uniqueness of non-negative solutions related to the critical eigenvalue problem associated with these equations. Additionally, I have investigated the Gelfand-type equation on Cartan-Hadamard manifolds, which involves classifying solutions based on their asymptotic behavior, stability, and intersection properties. The qualitative characteristics of these solutions can be significantly influenced by the global geometry of the underlying manifold, particularly with stochastic completeness. In that continuation, I also explore Dirichlet p-parabolicity, positivity-preserving properties, and the stochastic completeness of second-order operators on complete Riemannian manifolds.
Eigenvalue properties: Over time, my interest has expanded to the study of various eigenvalue properties associated with nonlinear operators in Euclidean space. Recently, I have become particularly focused on investigating the quantitative spectral stability of eigenvalues for specific types of domains under prescribed boundary conditions. In these problems, the sharp asymptotic behavior of eigenvalues with the perturbation parameter is expected to depend strongly on the vanishing order of the limit eigenfunction. The evaluation of the exact rate at which the eigenvalues of the perturbed problem converge to the eigenvalues of the limit problem can be studied through blow-up analysis. The spectral stability of singularly perturbed problems is relevant in several fields, including quantum mechanics, materials science, heat conduction, climate modeling, and acoustics.
Capacitary embedding theory: In the study of functional inequalities, embedding in the critical regime represents one of the most intriguing branches. Within this regime, a notable result is the exponential integrability, often referred to as the Moser-Trudinger inequality. My research interests extend to advancements in this area, particularly regarding trace exponential integrability, which depends on the dimension of integration. This regime is also embedded in the bounded oscillation spaces introduced by Brezis and Nirenberg. One can improve these embeddings to capture the fine properties of such functions through the introduction of a capacitary BMO space. These spaces enjoy an analog of the John-Nirenberg inequality, a result of fundamental importance in the study of functions of bounded mean oscillation. From this perspective, I recently started to explore a few Harmonic analysis tools. Specifically, I study maximal function estimates, bounded mean oscillation, vanishing mean oscillation, and the John-Nirenberg inequality concerning the dyadic Hausdorff contents adapted to cubes, the Choquet integrals defined in terms of these contents, and the function spaces generated by these integrals which were pioneered from the work of Adams.
The title of my thesis is “Study of the Poincaré-Hardy type Inequalities and eigenvalue problems for second-order elliptic PDEs". You can find my doctoral thesis here.