This is a continuing research of our previous work (S.-Y. A. Chang and J. Qing (1997),J. Funct. Anal.147, 327-362). In this paper we showW2,2-compactness of isospectral set within a subclass of conformal metrics, and discuss extremal properties of the zeta functional determinants, for certain elliptic boundary value problems on 4-manifolds with smooth boundary. To do so we establish some sharp Sobolev trace inequalities.
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