Singular Moser–Trudinger inequality on simply connected domains

ABSTRACT

In this paper the authors complete their study of the singularMoser-Trudinger embedding: in a previous result they have proven the existence of an extremal function for the singular Moser-Trudinger embedding

where α > 0 and β ∈ [0, 2) are such that and Ω ⊂ ℝ². This generalizes a well known result by Flucher, who has proven the case β = 0. This first proof is however far too technical and complicated for simply connected domains. Here we give a much simpler and more self-contained proof using complex analysis, which also generalizes the corresponding proof given by Flucher for such domains. This should make the previous work by the authors more easily accessible.

KEYWORDS:

• Concentration
• extremal function
• Moser-Trudinger embedding
• singular weight

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 35B38

Acknowledgments

The research work of the first author is supported by the Chilean Fondo Nacional de Desarrollo Científico y Tecnológico under the grant number FONDECYT Iniciación 11150017. The research work of the second author is supported by “Innovation in Science Pursuit for Inspired Research (INSPIRE)” under the IVR number 20140000099.