$ L^2 $-Gradient flows of spectral functionals

We study the $ L^2 $-gradient flow of functionals \mathscr F depending on eigenvalues Schrödinger potentials V for a wide class differential operators associated with closed, symmetric, and coercive bilinear forms, including case all Dirichlet forms (such as second order elliptic in Euclidean domains or Riemannian manifolds).We suppose that arises sum (-\theta) $-convex functional K proper domain {\mathbb K}\subset $, forcing admissible to stay above constant V_{\rm min} term {\mathscr{H}}(V) = \varphi(\lambda_1(V), \cdots, \lambda_ J(V)) which depends first J through {\mathrm C}^1 function \varphi $.Even though {\mathscr{H}} is not smooth perturbation convex (and it fact concave simple important cases eigenvalues) we do assume any compactness sublevels prove convergence Minimizing Movement method solution V\in H^1(0, T;L^2) inclusion V'(t)\in -\partial_L^- F(V(t)) under suitable compatibility conditions can be written as$ V'(t)+\sum\limits_{i 1}^ J\partial_i\varphi(\lambda_1(V(t)), \dots, J(V(t)))u_i^2(t)\in -\partial_F^- K(V(t)) $where (u_1(t), u_ J(t)) an orthonormal system eigenfunctions (\lambda_1(V(t)), , J(V(t))) \partial^-_L (resp. \partial^-_F $) denotes limiting Fréchet) subdifferential.