Affiliations: Institut Élie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine, Vandœuvre-lès-Nancy, France. E-mails: [email protected], E-mail: [email protected] | Aix-Marseille Université, CNRS, CPT UMR 7332, Marseille, France. E-mails: [email protected], [email protected] | Université de Toulon, CNRS, CPT UMR 7332, La Garde, France
Note: [] Corresponding author: Mourad Choulli, Institut Élie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine, B.P. 70239, 54506 Vandœuvre-lès-Nancy cedex, France. E-mail: [email protected]
Abstract: Let Ω be a C∞-smooth bounded domain of Rn, n≥1, and let the matrix a∈C∞(Ω¯;Rn2) be symmetric and uniformly elliptic. We consider the L2(Ω)-realization A of the operator −div (a∇·) with Dirichlet boundary conditions. We perturb A by some real valued potential V∈C0∞(Ω) and note AV=A+V. We compute the asymptotic expansion of tr (e−tAV−e−tA) as t↓0 for any matrix a with constant coefficients. In the particular case where A is the Dirichlet Laplacian in Ω, that is when a is the identity of Rn2, we make the four main terms appearing in the asymptotic expansion formula explicit and prove that L∞-bounded sets of isospectral potentials of A are bounded in H2(Ω).
