preprint

Inserted: 4 dec 2020

Year: 2019

Abstract:

We initiate the study of the higher order Escobar constants $I_k(M)$, $k\geq 3$, on bounded planar domains $M$. For a domain $M$ in $\mathbb{R}^2$ with Lipschitz and piecewise smooth boundary, we conjecture that its $k$-th Escobar constant $I_k(M)$ is bounded above by the $k$-th Escobar constant of the disk. This conjecture is answered in the affirmative for $M$ being a Euclidean $n$-gon and $k$ being greater or equal than $n$. For Euclidean and curvilinear polygons, we, in particular, provide bounds on $I_k(M)$ which depend only on the corner angles of this domain.