AbstractLet D be an integrally closed domain, $$\{V_{\alpha }\}$$
{
V
α
}
be the set of t-linked valuation overrings of D, and $$v_c$$
v
c
be the star operation on D defined by $$I^{v_c} = \bigcap _{\alpha } IV_{\alpha }$$
I
v
c
=
⋂
α
I
V
α
for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a $$v_c$$
v
c
-Noetherian domain if and only if D is a Krull domain, if and only if $$v_c = v$$
v
c
=
v
and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then $$v_c = v$$
v
c
=
v
if and only if D is a Dedekind domain.
The uniform convergence of the eigenfunctions expansions of the biharmonic operator in closed domain