Narrow Results

In this article, we have determined the remainder term for Hardy–Sobolev inequality in H¹(Ω) for Ω a bounded smooth domain and studied the existence, non existence and blow up of first eigen value and eigen function for the corresponding Hardy–Sobolev operator with Neumann boundary condition.

In this paper, we prove the long-time existence of the CR Yamabe flow on the compact strictly pseudoconvex CR manifold with positive CR invariant. We also prove the convergence of the CR Yamabe flow on the sphere by proving that: the contact form which is pointwise conformal to the standard contact form on the sphere converges exponentially to a contact form of constant pseudo-Hermitian sectional curvature. We also show that the eigenvalues of some geometric operators are non-decreasing under the unnormalized CR Yamabe flow provided that the pseudo-Hermitian scalar curvature satisfies certain conditions.

The purpose of this paper is two-fold. Firstly, we state and prove a Berezin–Li–Yau-type estimate for the sums of eigenvalues of , the fractional Laplacian operators restricted to a bounded domain Ω ⊂ ℝ^d for d ≥ 2 and α ∈ (0, 2]. Secondly, we provide an improvement to this estimate by using a pure analytical approach.

In this paper we will use variational methods and limiting approaches to give a complete solution to the maximization problems of the Dirichlet/Neumann eigenvalues of the one-dimensional p-Laplacian with integrable potentials of fixed L¹-norm.

A key basis for seeking solutions of the Camassa–Holm equation is to understand the associated spectral problem