$L^2$-Gradient Flows of Spectral Functionals

TL;DR

This paper investigates the $L^2$-gradient flow of spectral functionals related to eigenvalues of Schrödinger operators, proving convergence of a variational scheme even with non-convex and non-smooth functionals.

Contribution

It establishes the convergence of the Minimizing Movement method for a broad class of spectral functionals, including non-convex and non-smooth cases, without requiring sublevel compactness.

Findings

Proves convergence of the Minimizing Movement scheme to a solution of the gradient flow.

Derives a differential inclusion involving eigenvalues and eigenfunctions.

Handles non-convex spectral functionals with minimal regularity assumptions.

Abstract

We study the $L^2$-gradient flow of functionals $\mathcal F$ depending on the eigenvalues of Schr\"odinger potentials $V$ for a wide class of differential operators associated to closed, symmetric, and coercive bilinear forms, including the case of all the Dirichlet forms (as for second order elliptic operators in Euclidean domains or Riemannian manifolds). We suppose that $\mathcal F$ arises as the sum of a $-\theta$-convex functional $\mathcal K$ with proper domain $\mathbb{K}\subset L^2$ forcing the admissible potentials to stay above a constant $V_{\rm min}$ and a term $\mathcal H(V)=\varphi(\lambda_1(V),\cdots,\lambda_J(V))$ which depends on the first $J$ eigenvalues associated to $V$ through a $C^1$ function $\varphi$. Even if $\mathcal H$ 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 $\mathcal K$, we prove the convergence of the Minimizing Movement method to a solution $V\in H^1(0,T;L^2)$ of the differential inclusion $V'(t)\in -\partial_L^-\mathcal F(V(t))$, which under suitable compatibility conditions on $\varphi$ can be written as \[ V'(t)+\sum_{i=1}^J\partial_i\varphi(\lambda_1(V(t)),\dots, \lambda_J(V(t)))u_i^2(t)\in -\partial_F^-\mathcal K(V(t)) \] where $(u_1(t),\dots, u_J(t))$ is an orthonormal system of eigenfunctions associated to the eigenvalues $(\lambda_1(V(t)), ,\dots,\lambda_J(V(t)))$.