This work develops a general variational–spectral framework for functionals of the form F (u) = ||R (u) ||², where R is a nonlinear operator between Hilbert spaces and critical points are characterized by the condition R (u₀) = 0. A fundamental structural identity shows that the second variation at a critical point is given by Hess F (u₀) = 2 L*L, where L = DR (u₀) is the linearisation of the residual. This places the analysis within the theory of positive self-adjoint operators of the form L*L. Under standard assumptions — including ellipticity, self-adjointness, and the existence of a spectral gap — we establish a local bilateral coercivity estimate c ||h||²H² ≤ F (u₀ + h) ≤ C ||h||²H², showing that the functional is equivalent to the H²-norm in a neighbourhood of a non-degenerate critical point. The proof combines elliptic regularity, spectral theory, and nonlinear remainder estimates controlled by Sobolev embeddings. The framework is applied to geometric problems in dimensions two and four. In the two-dimensional case, the theory recovers curvature-based functionals in conformal geometry, where the stability operator is a Schrödinger-type Jacobi operator. In dimension four, the analysis applies to deviation functionals associated with Einstein metrics, where the linearisation involves the Lichnerowicz Laplacian and coercivity holds under a spectral gap condition in the transverse–traceless sector. The associated gradient flow ∂ₜu = −L*L u defines a fourth-order dissipative evolution whose relaxation rate is governed by the square of the lowest eigenvalue. This identifies a common variational–spectral mechanism underlying stability, rigidity, and dissipative dynamics across a class of geometric problems.
Mário César Garms Thimoteo (Mon,) studied this question.