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Optimal Sobolev and Hardy-Rellich constants under Navier boundary conditions

by Gazzola, F., Grunau, H.-Ch., Sweers, G..

Series: 2009-17, Preprints

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators

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 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.

optimal constant, Sobolev embedding, Hardy-Rellich inequality