SOBOLEV TRACE INEQUALITIES OF ORDER FOUR

Abstract

We establish sharp trace Sobolev inequalities of order four on Euclidean d-balls for d >= 4. When d = 4, our inequality generalizes the classical second-order LebedevMilin 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.