Sharp Moser-Trudinger inequalities for the Laplacian without boundary conditions

TL;DR

This paper establishes a sharp Moser-Trudinger inequality for the Laplacian on the unit ball without boundary conditions, extending known results to a broader setting and introducing new integral representation techniques.

Contribution

The authors derive the first sharp Moser-Trudinger inequalities for the Laplacian on the ball without boundary conditions, using novel integral representations and asymptotic kernel estimates.

Findings

Established sharp inequalities for W^{2,n/2}(B_n) without boundary conditions.

Extended results to inequalities with general Borel measures and trace type.

Provided new integral formulas for solutions of the Poisson equation on the ball.

Abstract

We derive a sharp Moser-Trudinger inequality for the borderline Sobolev imbedding of W^{2,n/2}(B_n) into the exponential class, where B_n is the unit ball of R^n. The corresponding sharp results for the spaces W_0^{d,n/d}(\Omega) are well known, for general domains \Omega, and are due to Moser and Adams. When the zero boundary condition is removed the only known results are for d=1 and are due to Chang-Yang, Cianchi and Leckband. Our proof is based on general abstract results recently obtained by the authors, and on a new integral representation formula for the "canonical" solution of the Poisson equation on the ball, that is the unique solution of the equation \Delta u=f which is orthogonal to the harmonic functions on the ball. The main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation, expressed in terms of its distribution function. We will also consider the situation where the exponential class is endowed with more general Borel measures, and obtain corresponding sharp Moser-Trudinger inequalities of trace type.