Symmetry properties for the extremals of the Sobolev trace embedding

In this article we study symmetry properties of the extremals for the Sobolev trace embedding H¹(B(0,μ))L^q(∂B(0,μ)) with 1q2(N−1)/(N−2) for different values of μ. These extremals u are solutions of the problem

We find that, for 1q<2(N−1)/(N−2), there exists a unique normalized extremal u, which is positive and has to be radial, for μ small enough. For the critical case, q=2(N−1)/(N−2), as a consequence of the symmetry properties for small balls, we conclude the existence of radial extremals. Finally, for 1<q2, we show that a radial extremal exists for every ball.