Existence of global minimizers for compactly perturbed coercive functionals in Hilbert Spaces: Spectral thresholds and gradient flow stability

This work studies variational energy functionals defined on separable Hilbert spaces of the form E (s) =λS (s) −⟨Bs, s⟩E (s) = S (s) - Bs, s (s) =λS (s) −⟨Bs, s⟩, where SSS is a strongly convex coercive functional and BBB is a compact self-adjoint operator. Using the direct method of the calculus of variations, we establish explicit spectral conditions guaranteeing the existence of global minimizers. We further analyze local stability properties and derive conditions for strong monotonicity of the associated gradient operator. Under the stronger spectral bound λα>2∥B∥ > 2\|B\|λα>2∥B∥, the gradient flow generated by the functional is shown to be globally well-posed and exponentially stable via nonlinear semigroup theory. Sharpness of the thresholds is demonstrated through explicit counterexamples in ℓ2²ℓ2. The results provide a rigorous framework for analyzing compact perturbations of coercive variational structures in infinite-dimensional spaces.