Publications and Preprints

Abstract

We establish Lyapunov-type inequality for equations concerning general class of second order non-symmetric elliptic operators with singular coefficients. Our approach is based on the probabilistic representation of solutions and stochastic calculus. We also discuss a Lyapunov-type inequality for equations pertaining to second order symmetric operator with some regularity assumptions on the coefficients and a nonlinear Neumann boundary condition.

Abstract

We review briefly the fundamental solutions to some of the most important partial differential operators, which are well available in the literature. These are very crucial in analysis and partial differential equations (PDEs). Among several applications, these are used, for instance, in studying regularity and growth of solutions.

Abstract

We establish a Lyapunov-type inequality for a class of mixed local-nonlocal operator. We employ solution representation formula for the associated boundary value problem. Furthermore, as an application of the Lyapunov-type inequality, we show a positive lower bound for the generalized principal eigenvalue of the operator.  

Abstract

We investigate the boundary regularity of solutions to a class of variable-exponent gradient degenerate mixed fully nonlinear local-nonlocal elliptic Dirichlet problems. A crucial feature of the operators under consideration is that they degenerate on the set of critical points, {x: Du(x)=0}. First, we establish the Lipschitz regularity of solutions using the Ishii-Lions viscosity method when the order of the fractional Laplacian, 1/2<s<1, under general conditions. Due to the inapplicability of the comparison principle for the equations under consideration, in general, the classical Perron's method for the existence of a solution can not employed. However, utilizing the Lipschitz estimates and vanishing viscosity method, we prove the existence of a solution. Additionally, we prove interior $C^{1,\delta}$ regularity of viscosity solutions using an improvement of the flatness technique when s is close enough to 1. Furthermore, under suitable assumptions, we establish the H\"older regularity of solutions up to the boundary.

 Abstract

This paper investigates a gradient-degenerate, nonlocal version of the generalized $p$-Laplacian introduced by Baravdish, Cheng, Svensson, and \r{A}str\"om \cite{Barav 2}. A key feature of this operator is its degeneracy along the set of critical points, which prevents the application of standard comparison principles. We establish the existence and interior Lipschitz regularity of viscosity solutions by employing an adapted Ishii-Lions ``doubling variables" technique. We also identify a setting in which uniqueness of solutions is proved.

Abstract

 We investigate a class of equations involving fully nonlinear degenerate elliptic operators with a Hamiltonian term. A distinctive feature of this class is the presence of gradient degeneracy of variable-exponent double phase type. We first prove a comparison principle for viscosity subsolutions and supersolutions. Then, using the Ishii-Lions lemma, we establish an interior H\"older regularity result. To handle the gradient degeneracy, we adopt the techniques introduced by Imbert and Silvestre (Advances in Mathematics, 233(1), (2013), 196--206). Furthermore, under suitable conditions, we extend the H\"older regularity up to the boundary.

In this survey article, we recount some of the most important developments on the Hot Spots conjecture available. It says that any eigenfunction corresponding to the first non-zero eigenvalue of Neumann Laplacian in a bounded smooth domain attains its maximum and minimum at the boundary. We review several well-known findings concerning specific domains, for which the conjecture holds or fails.