$ L^2 $-Gradient flows of spectral functionals

We study the $ L^2 $-gradient flow of functionals $ \mathscr F $ depending on the eigenvalues of Schrödinger potentials $ V $ for a wide class of differential operators associated with closed, symmetric, and coercive bilinear forms, including the case of all the Dirichlet forms (such as for second order elliptic operators in Euclidean domains or Riemannian manifolds).

We suppose that $ \mathscr F $ arises as the sum of a $ (-\theta) $-convex functional $ \mathscr K $ with proper domain $ {\mathbb K}\subset L^2 $, forcing the admissible potentials to stay above a constant $ V_{\rm min} $, and a term $ {\mathscr{H}}(V) = \varphi(\lambda_1(V), \cdots, \lambda_ J(V)) $ which depends on the first $ J $ eigenvalues associated with $ V $ through a $ {\mathrm C}^1 $ function $ \varphi $.

Even though $ {\mathscr{H}} $ is not a smooth perturbation of a convex functional (and it is in fact concave in simple important cases as the sum of the first $ J $ eigenvalues) and we do not assume any compactness of the sublevels of $ \mathscr K $, we prove the convergence of the Minimizing Movement method to a solution $ V\in H^1(0, T;L^2) $ of the differential inclusion $ V'(t)\in -\partial_L^- \mathscr F(V(t)) $, which under suitable compatibility conditions on $ \varphi $ can be written as

$ V'(t)+\sum\limits_{i = 1}^ J\partial_i\varphi(\lambda_1(V(t)), \dots, \lambda_ J(V(t)))u_i^2(t)\in -\partial_F^- \mathscr K(V(t)) $

where $ (u_1(t), \dots, u_ J(t)) $ is an orthonormal system of eigenfunctions associated with the eigenvalues $ (\lambda_1(V(t)), , \dots, \lambda_ J(V(t))) $ and $ \partial^-_L $ (resp. $ \partial^-_F $) denotes the limiting (resp. Fréchet) subdifferential.