Functional inequalities such as Hardy inequality, Rellich inequality, logarithmic Sobolev inequality serve as a major tool in partial differential equations, operator theory, spectral theory, and mathematical physics.  I am interested in the following aspects of these functional inequalities:

   (i) Characterizing (or identifying) admissible spaces of weight functions in these inequalities

   (ii) Existence of minimizer associated with these functional inequalities

   (iii) Finding "Optimal weight-functions" in these inequalities

   (iv) Quantitative stability of functional inequalities

Such problems are related to the Shape optimization problems. One can derive crucial information of a mathematical model of certain physical phenomena by studying an appropriate optimization problem related to the eigenvalues of the underlying elliptic operators. Hence the existence of optimizers is a natural question that arises in investigating such models. Also, it is important and interesting to analyze the qualitative properties of the optimizers of these optimization problems when they exist.

Unique continuation at infinity (UCI) is one of the classical research topics with widespread applications in various areas. Formally, UCI for a solution u of a partial differential equation (PDE) concerns the decay behavior of u and assures whether u ≡ 0 in R^d if it decays at infinity faster than a certain threshold. In this context, I am particularly interested in Landis-type unique continuation results, which are related to addressing a well-known conjecture in this topic proposed by E.M. Landis.  

I have a keen interest in the analysis on discrete settings. This includes studying various aspects of functional inequalities (Hardy, Rellich, log-Sobolev inequality, etc.) on discrete graph, unique continuation properties for certain discrete equations, their non-local analogue, and certain phenomena related to discrete eigenvalue problems (e.g., asymptotic behaviour of coupling constant).

I am interested in studying the existence and multiplicity results of the following type of PDEs:

  (i) Brezis-Nirenberg type problems: PDEs with critical Sobolev exponent

  (ii) Positone/Semipositone problems: Existence and multiplicity of positive solutions to non-linear boundary value problems with certain growth conditions on the non-linearity.