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 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.
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