Singular Moser–Trudinger inequality on simply connected domains

Abstract

≤ 1, and Ω ⊂ R 2 . This generalizes a well known result by Flucher, who has proven the case β = 0. The proof in [CR] 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 [CR] more easily accessible.

Extremal functions for singular Moser–Trudinger embeddings

“…and sup u∈H 1 0 (Ω), Ω |∇u| 2 dx≤1 Ω |x| 2α e βu 2 dx = +∞, (12) for any β > 4π(1 + α). Existence of extremals for (11) has recently been proved in [7] and [8]. The strategy is similar to the one used for the case α = 0.…”

On attainability of Moser-Trudinger inequality with logarithmic weights in higher dimensions

“…Moser-Trudinger type inequality had been an interesting topic of research for several authors. For Moser-Trudinger inequality with singular weight we refer to [7,8,14,19,27]. For the study of this inequality in entire space and on manifolds we refer to [17,23,29,33,35].…”

On the isoperimetric problem with perimeter density <inline-formula><tex-math id="M1">\begin{document} $r^p$ \end{document}</tex-math></inline-formula>

“…The motivation for studying negative values of p and adding these additional assumptions come from the singular Moser-Trudinger functional [2]. These assumptions arise naturally in the harmonic transplantation method of Flucher [18] (see [12] and [13]) to establish the existence of extremal functions for the singular Moser-Trudinger functional. Without going into the details, the connection is the following: one uses (1) for the level sets of the Greens function G Ω,0 with singularity at 0.…”