1 research outputs found
Heat trace asymptotics and boundedness in the second order Sobolev space of isospectral potentials for the Dirichlet Laplacian
International audienceLet be a -smooth bounded domain of , , and let the matrix be symmetric and uniformly elliptic. We consider the -realization of the operator -\mydiv ( {\bf a} \nabla \cdot) with Dirichlet boundary conditions. We perturb by some real valued potential and note . We compute the asymptotic expansion of \mbox{tr}\left( e^{-t A_V}-e^{-t A}\right) as for any matrix whose coefficients are homogeneous of degree . In the particular case where is the Dirichlet Laplacian in , that is when is the identity of , we make the four main terms appearing in the asymptotic expansion formula explicit and prove that -bounded sets of isospectral potentials of are -compact for