A NOTE ON THE SOBOLEV TRACE INEQUALITY

초록

Consider the classical Sobolev trace inequality parallel to del phi parallel to(L2(R+n)) >= K parallel to phi parallel to(L2(n-1)/n-2(partial derivative R+n)) for all phi is an element of W-0(1,2)(R-+(n)), where K is the best constant. Here, W-0(1,2)(R-+(n)) is the space obtained by taking the completion in the norm parallel to del phi parallel to(L2(R+n)) of the set of all smooth functions with support contained in the closure of R-+(n), and n >= 3. Let M be the set of functions for which we have equality in the Sobolev trace inequality above. In this note, we show that there is a positive constant a such that parallel to del phi parallel to(2)(L2(R+n)) - K-2 parallel to phi parallel to(2)(L2(n-1)/Ln-2(partial derivative R+n)) >= alpha d(phi, M)(2) for all phi is an element of W-0(1,2)(R-+(n)), where d is the distance in the Sobolev space W-0(1,2)(R-+(n)).

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