Jiuyi Zhu

Improved Moser-Trudinger Inequality Involving Lp Norm in n Dimensions

Abstract The paper is concerned about an improvement of Moser-Trudinger inequality involving Lp norm for a bounded domain in n dimensions. Let be the first eigenvalue associated with n-Laplacian. We obtain the following strengthened Moser-Trudinger inequality with blow-up analysis for 0 ≤ α < λ̅(Ω) and 1 < p ≤ n, and the supremum is infinity for α ≥ λ̅(Ω), where and ωn−1 is the surface area of the unit ball in ℝn. We also obtain the existence of the extremal functions for (0.2).

Doubling Property and Vanishing Order of Steklov Eigenfunctions

The paper is concerned with the doubling estimates and vanishing order of the Steklov eigenfunctions on the boundary of a smooth domain in ℝ n . The eigenfunction is given by a Dirichlet-to-Neumann map. We improve the doubling property shown by Bellova and Lin. Furthermore, we show that the optimal vanishing order of Steklov eigenfunction is everywhere less than Cλ where λ is the Steklov eigenvalue and C depends only on Ω.

Liouville-type Theorems for Fully Nonlinear Elliptic Equations and Systems in Half Spaces

Abstract The main purpose of this paper is to establish Liouville-type theorems and decay estimates for viscosity solutions to a class of fully nonlinear elliptic equations or systems in half spaces without the boundedness assumptions on the solutions. Using the blow-up method and doubling lemma of [18], we remove the boundedness assumption on solutions which was often required in the proof of Liouville-type theorems in the literature. We also prove the Liouville-type theorems for supersolutions of a system of fully nonlinear equations with Pucci extremal operators in half spaces. Liouville theorems and decay estimates for high order elliptic equations and systems have also been established by the authors in an earlier work [15] when no boundedness assumption was given on the solutions.