L2-Gradient flows of spectral functionals

We study the L-2-gradient flow of functionals F depending on the eigenvalues of Schro spacing diaeresis dinger 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 F arises as the sum of a (-theta)-convex functional K with proper domain K subset of L-2, forcing the admissible potentials to stay above a con-stant Vmin, and a termH (V) = phi(lambda 1(V), middot middot middot, lambda J(V)) which depends on the first J eigenvalues associated with V through a C-1 function phi.Even though 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 ei-genvalues) and we do not assume any compactness of the sublevels of K , we prove the convergence of the Minimizing Movement method to a solution V is an element of H-1(0, T; L-2) of the differential inclusion V'(t) is an element of- partial differential L-F(V (t)), which under suitable compatibility conditions on phi can be written asJ V (1)(t) + partial differential i phi(lambda(1)(V (t)), ... ,lambda J(V(t)))u(i)(2) (t) is an element of - partial differential F -K (V (t)) i=1where (u1(t),..., uJ(t)) is an orthonormal system of eigenfunctions associated with the eigenvalues (lambda(1)(V (t)), , ... , lambda J(V(t))) and partial differential L- (resp. partial differential F- ) denotes the limiting (resp. Fre acute accent chet) subdifferential.