On the heat kernel of a class of fourth order operators in two dimensions: sharp Gaussian estimates and short time asymptotics

摘要

We consider a class of fourth order uniformly elliptic operators in planarEuclidean domains and study the associated heat kernel. For operators with$L^{\infty}$ coefficients we obtain Gaussian estimates with best constants,while for operators with constant coefficients we obtain short time asymptoticestimates. The novelty of this work is that we do not assume that theassociated symbol is strongly convex. The short time asymptotics reveal abehavior which is qualitatively different from that of the strongly convexcase.