Abstract
We prove that the best constant for the critical embedding of higher order Sobolev spaces does not depend on all the traces. The proof uses a comparison principle due to Talenti (Ann Scuola Norm Sup Pisa Cl Sci 3(4): 697–718, 1976) and an extension argument which enables us to extend radial functions from the ball to the whole space with no increase of the Dirichlet norm. Similar arguments may also be used to prove the very same result for Hardy-Rellich inequalities.
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Gazzola, F., Grunau, HC. & Sweers, G. Optimal Sobolev and Hardy–Rellich constants under Navier boundary conditions. Annali di Matematica 189, 475–486 (2010). https://doi.org/10.1007/s10231-009-0118-5
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DOI: https://doi.org/10.1007/s10231-009-0118-5