Heat Trace Asymptotics and Compactness of Isospectral Potentials for the Dirichlet Laplacian

Let Ω be a C∞-smooth bounded domain of R, n ≥ 1, and let the matrix a ∈ C(Ω;R 2 ) be symmetric and uniformly elliptic. We consider the L(Ω)-realization A of the operator −div(a∇·) with Dirichlet boundary conditions. We perturb A by some real valued potential V ∈ C∞ 0 (Ω) and note AV = A + V . We compute the asymptotic expansion of tr ( eV − e ) as t ↓ 0 for any matrix a whose coefficients are homogeneous of degree 0. In the particular case where A is the Dirichlet Laplacian in Ω, that is when a is the identity of R 2 , 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 H-compact for s < 2.