Heat trace asymptotics and compactness of isospectral potentials for the Dirichlet Laplacian

Abstract

Let $\Omega$ be a $C^{\infty }$-smooth bounded domain of ${\mathbb{R}}^n$, $n\ge 1$, and let the matrix $\mathbf{a}\in C^{\infty }(\overline{\Omega };{\mathbb{R}}^{n^2})$ be symmetric and uniformly elliptic. We consider the $L^2(\Omega )$-realization $A$ of the operator $-\mydiv ( {\bf a} \nabla \cdot)$ with Dirichlet boundary conditions. We perturb $A$ by some real valued potential $V\in C_0^{\infty }(\Omega )$ and note $A_V=A+V$. We compute the asymptotic expansion of $\mbox{tr}\left( e^{-t A_V}-e^{-t A}\right)$ as $t\downarrow 0$ for any matrix $\mathbf{a}$ whose coefficients are homogeneous of degree $0$. In the particular case where $A$ is the Dirichlet Laplacian in $\Omega$, that is when $\mathbf{a}$ is the identity of ${\mathbb{R}}^{n^2}$, we make the four main terms appearing in the asymptotic expansion formula explicit and prove that $L^{\infty }$-bounded sets of isospectral potentials of $A$ are $H^s$-compact for $s<2$.