Barbatis G., Davies E.B.
Sharp bounds on heat kernels of higher order uniformly elliptic operators.
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ABSTRACT. We study higher order uniformly elliptic operators with measurable highest
order coefficients acting on Euclidean domains. We obtain Gaussian heat
kernel bounds for such operators and establish explicit estimates for the
constant in the exponential term. These are expressed in terms of the
ellipticity ratio of the operator and are sharp for powers of the Laplacian.
We consider separately the case of homogeneous and non-homogeneous operators
and distinguish between short- and long-time estimates. In each case, all
our estimates are sharp in an appropriate asymptotic sense.