My research has been in the analysis of Elliptic PDE




Doctoral Research

In the following, I briefly describe some of my doctoral research areas:

Generalized Hardy-Sobolev inequlaity and it’s application: A significant portion of my thesis focuses on the exploration of the Hardy-Sobolev inequality, which is widely recognized in the field of Partial Differential Equations and related areas. This inequality establishes that, for a smooth function f with compact support, the L^p norm of f divided by |x| is bounded by the L^p norm of the gradient of f. Numerous research have been conducted to enhance this inequality in various aspects, resulting in a substantial body of literature. A recommended resource for understanding the origins of the Hardy inequality and its associated outcomes is the monograph titled The Hardy Inequality. About Its History and Some Related Results.

My particular interest lies in generalizing this inequality by incorporating optimal weights within different function spaces, such as Lorentz space, Lorentz-Zygmund space, and Orlicz space. These are multi parameter family of function spaces that refine the classical Lebesgue space. To effectively investigate these rearrangement invariant function spaces, a fundamental concept to grasp is the symmetrization of a function. Notably, there exist well-known inequalities involving symmetrization, including the Hardy-Littlewood inequality and Pólya–Szegő inequality. Some useful references for this topic are the following:

One good motivation to generalize the inequality in this direction is that the weight functions in the classical Hardy inequality lie in Lorentz and Lorentz-Zygmund spaces (depending on the dimension). My thesis encompasses two primary objectives. The first part investigates three distinct variations of weighted Hardy-Sobolev inequalities, with the aim of identifying the optimal function spaces for each type of inequality. Two of these variants consider scenarios where the test functions do not vanish on the boundary of the domain. In addition to rearrangement invariant function spaces, radial and cylindrical function spaces are employed in the analysis.

The second part of my thesis focuses on identifying function spaces in which the optimal constant of the weighted Hardy-Sobolev inequality is achieved. We present a general sufficient condition on the weight function that guarantees the attainment of the best constant. Specifically, the attainment of this constant is equivalent to finding a weak solution to the corresponding weighted eigenvalue problems for the p-Laplace operator, considering zero Dirichlet, Neumann, and Steklov boundary conditions. As a by-product, we explore weighted eigenvalue problems within a broad class of weight functions. This problem is studied concerning the principal eigenvalue and subsequently for an infinite set of eigenvalues.

Bifurcation of eigenvalues: The complete spectrum of the p-Laplace operator is not known. But it is well known that the spectrum consists of an increasing sequence of eigenvalues tending to infinity. An eigenvalue`a’ of the p-Laplace operator is called a bifurcation point (from zero) if there exists a sequence of pairs (a_n, v_n) of the perturbed eigenvalue problem such that the eigenvalues (a_n) converges to `a’, and the associated weak solutions (v_n) converges to 0 in the appropriate Sobolev space. Extensive research has been dedicated to investigate the existence of bifurcation points for the p-Laplace operator with Dirichlet boundary conditions. It has been established that the first positive Dirichlet eigenvalue serves as the bifurcation point for the p-Laplace operator. In 1971, Paul Rabinowitz made a notable contribution by demonstrating an interesting phenomenon applicable to a broad class of elliptic problems. This phenomenon states that there exists an uncountable set of non-trivial weak solutions that bifurcate from the point (a,0), whereas the set is either unbounded or converges to the point (b,0), where 'b' represents another eigenvalue of the p-Laplace operator. Furthermore, depending upon the multiplicity of 'a', this continuum can be decomposed into two subcontinua of non-trivial solutions.

The objective of my research is to investigate the corresponding outcomes for the weighted eigenvalue problem involving the p-Laplace operator. Specifically, we aim to explore this problem in relation to Neumann and Steklov boundary conditions, as well as general weight functions within the Lorentz and Lorentz-Zygmund spaces. Our findings indicate that the first positive eigenvalue serves as the bifurcation point for the perturbed eigenvalue problem, and the bifurcating set follows the properties similar to what was observed in Rabinowitz's paper. One of the most important tool in this analysis is the topological degree theory (for example, Leray–Schauder degree) for certain class of maps defined on the solution space. Some useful references for this topic are as follows:



Postdoctoral Research

My post doctoral research includes the following topics:

(a) Optimization of the first Dirichlet eigenvalue over certain rearrangement invariant classes.

(b) Positivity of the solutions on the entire R^N for some semipositone problems.

(c) Shape optimization of the fractional eigenvalues.

(d) The study of spectrum for inhomogeneous fractional operators.

(e) Regularity theory for inhomogeneous elliptic operators.

(f) Global compactness results for nonlinear nonlocal systems.



Preprints

Publications