TWO OBSTRUCTIONS TO COERCIVITY: A RADICAL-QUOTIENT CONSTRUCTION FOR PSD BILINEAR FORMS

In standard variational practice, a symmetric positive semidefinite bilinear form is chosen a priori, and its subspace of free directions is subsequently determined as its radical. This note formalizes the inverse procedure. The construction has five steps of three types—two free data (L and g), one canonical construction (the quotient V /L), and two operations (assembly and level separation)—organized around the bijection between symmetric PSD forms with radical L and positive definite forms on V /L. The resulting diagnostic separates the failure of coercivity into two exhaustive obstructions: the seminorm does not vanish on L, or the induced form on V /L does not dominate the quotient seminorm. The two obstructions correspond to the two decisions that the standard order conflates: the choice of the free subspace (modeling) and the choice of the metric on the quotient (analysis). Keywords: bilinear forms; radical; quotient spaces; construction; variational methods; coercivity; two-obstruction criterion; variational stability MSC 2020: 15A63, 15A21, 15A03, 47A07, 65N30, 65N12