We establish sharp trace Sobolev inequalities of order four on Euclidean -balls for . When , our inequality generalizes the classical second-order Lebedev–Milin inequality on Euclidean -balls. Our method relies on the use of scattering theory on hyperbolic -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 -balls, which surprisingly is not the flat metric on the ball.
Antonio G. Ache. Sun-Yung Alice Chang. "Sobolev trace inequalities of order four." Duke Math. J. 166 (14) 2719 - 2748, 1 October 2017. https://doi.org/10.1215/00127094-2017-0014