Semicoercivity

Semicoercivity is a property of bilinear forms in functional analysis that lies between coercivity and non-coercivity. It arises in variational formulations where the energy functional controls the norm only in certain directions, while directions in a finite-dimensional subspace may contribute no energy.

Formally, let V be a real or complex Hilbert space and a: V × V → R a continuous bilinear form. a is semicoercive if there exists a finite-dimensional subspace Z ⊂ V such that V = V0 ⊕ Z for some closed subspace V0 and there is a constant α > 0 with a(u,u) ≥ α ||u||^2 for all u ∈ V0. Equivalently, there exists α > 0 and a continuous seminorm p on V for which a(u,u) ≥ α ||u||^2 − p(u)^2 for all u ∈ V. In this sense the form is coercive on the subspace V0, while directions in Z do not contribute to the energy.

Semicoercivity commonly appears when the energy vanishes along a finite-dimensional set of rigid or neutral motions. A typical instance is in elasticity, where the internal energy is zero on rigid body motions, or in problems with free boundaries that permit nontrivial zero-energy directions. The concept is also used in the analysis of variational inequalities and saddle-point problems where coercivity fails on the whole space but holds on a complementary subspace.

Consequences include existence results for the variational problem a(u, v) = f(v) and, in general, non-uniqueness up to elements of Z. To obtain uniqueness, one supplements the formulation with additional constraints that fix the finite-dimensional neutral directions, or imposes compatibility conditions on the right-hand side f with respect to Z. Semicoercivity thus provides a practical framework for solvability when strict coercivity is absent.