A threshold phenomenon for embeddings of H0mH^m_0 into Orlicz spaces

We consider a sequence of positive smooth critical points of the Adams-Moser-Trudinger embedding of $H^m_0$ into Orlicz spaces. We study its concentration-compactness behavior and show that if the sequence is not precompact, then the liminf of the $H^m_0$-norms of the functions is greater than or equal to a positive geometric constant.Comment: 14 Page

The substrate lends a hand

Duramycin is a small post-translationally modified peptide with antibody-like affinity for phosphatidylethanolamine. As it turns out, the same functionality that is essential for duramycin activity helps to catalyze the formation of its conformationally constrained and compact polycyclic architecture

A classification theorem for Helfrich surfaces

In this paper we study the functional \SW_{\lambda_1,\lambda_2}, which is the the sum of the Willmore energy, $\lambda_1$-weighted surface area, and $\lambda_2$-weighted volume, for surfaces immersed in $\R^3$. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our main result is a complete classification of all smooth immersed critical points of the functional with $\lambda_1\ge0$ and small $L^2$ norm of tracefree curvature. In particular we prove the non-existence of critical points of the functional for which the surface area and enclosed volume are positively weighted.Comment: 21 page