Sobolev trace inequalities of order four

Abstract

We establish sharp trace Sobolev inequalities of order four on Euclidean $d$-balls for $d\ge 4$. When $d=4$, our inequality generalizes the classical second-order Lebedev–Milin inequality on Euclidean $2$-balls. Our method relies on the use of scattering theory on hyperbolic $d$-balls. As an application, we characterize the extremal metric of the main term in the log-determinant formula corresponding to the conformal Laplacian coupled with the boundary Robin operator on Euclidean $4$-balls, which surprisingly is not the flat metric on the ball.