quasicoercive

Quasi-coercive, also written quasicoercive, is a term used in functional analysis and the theory of partial differential equations to describe a relaxed form of coercivity for a bilinear form or operator. Let V be a space with a continuous embedding into a larger Hilbert space H, and consider a continuous bilinear form a: V × V → R. The form is called quasi-coercive if there exist constants α > 0 and β ≥ 0 such that for all v in V, a(v, v) ≥ α ||v||_V^2 − β ||v||_H^2. When β = 0, the form is coercive.

This inequality is a version of Garding's inequality and expresses that the form is bounded below by a positive multiple of the V-norm up to a lower-order term controlled by the H-norm. Quasi-coercivity commonly arises in elliptic problems with lower-order terms, where the leading part provides coercivity but additional terms prevent full coercivity.

Implications and use: Quasi-coercivity implies that the associated operator is bounded below modulo a compact perturbation, making it a Fredholm operator of index zero under suitable conditions. This underpins well-posedness results for variational problems and supports the stability and convergence of Galerkin and other discretization methods. In practice, one often handles quasi-coercive forms by shifting or bounding techniques that reduce the problem to a coercive one or by appealing to the Lax–Milgram framework on a modified inequality.

See also Garding inequality, coercivity, Lax–Milgram theorem, and Fredholm theory.