论文标题
$ l^2 $ - 频谱功能的率流
$L^2$-Gradient Flows of Spectral Functionals
论文作者
论文摘要
We study the $L^2$-gradient flow of functionals $\mathcal F$ depending on the eigenvalues of Schrödinger 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). 我们认为$ \ Mathcal f $作为$-θ$ -Convex函数$ \ Mathcal k $的总和，具有适当的域$ \ Mathbb {k} \ subset l^2 $强迫可接受的潜力，以保持不变的$ v _ h（v）=φ（λ_1（v），\ cdots，λ_j（v））$，取决于第一个$ j $ eigenvalues通过$ c^1 $函数$φ$与$ v $相关的$ j $ eigenValues。 即使$ \数学h $也不是凸功能的平稳扰动（实际上，在简单的重要情况下，它是第一个$ j $ eigenvalues的总和），并且我们不假定$ \ \ \ \级k $的sublevels sublesents n Mathcal k $ sublevel，我们证明了$ $ v v f的最小值$ v^l^1（n^n hh^l^n^n^n^l^1 f^n^n^n^n n^n n^n n n^l^1（h）^1（h）^1（^1）包含$ v'（t）\ in - \ partial_l^ - \ Mathcal f（v（t））$，在$φ$上适当的兼容条件下，可以将其写入\ [v'（v'（t）+\ sum_ {i = 1}^j \ j \ partial_i或 - \ partial_f^ - \ Mathcal k（v（t））\]其中$（u_1（t），\ dots，u_j（t））$是与eigenvalues $ $（λ_1（v（v（t）），，\ dots，λ_j（v（t）），$（λ_1（v（t））$）$（λ_1（v（t））$）$（v（t））的正态系统。
We study the $L^2$-gradient flow of functionals $\mathcal F$ depending on the eigenvalues of Schrödinger 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 $-θ$-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 $\mathcal H(V)=φ(λ_1(V),\cdots,λ_J(V))$ which depends on the first $J$ eigenvalues associated to $V$ through a $C^1$ function $φ$. 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 $φ$ can be written as \[ V'(t)+\sum_{i=1}^J\partial_iφ(λ_1(V(t)),\dots, λ_J(V(t)))u_i^2(t)\in -\partial_F^-\mathcal K(V(t)) \] where $(u_1(t),\dots, u_J(t))$ is an orthonormal system of eigenfunctions associated to the eigenvalues $(λ_1(V(t)), ,\dots,λ_J(V(t)))$.
