Narrow Results

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space $H_0^m(\mathrm{\Omega })$, where $\mathrm{\Omega }$ is any bounded, smooth, open subset of ${ℝ}^{2m}$, $m\ge 1$. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

In this paper, we study eigenvalues of poly-Laplacian and fractional Laplacian on bounded domains in Euclidean space. In particular, we focus all attentions on lower bounds for sum of eigenvalues of poly-Laplacian and fractional Laplacian. In addition, for the case of Laplacian, we also discuss lower bounds of eigenvalues of Dirichlet problem on a polytope.