Gaussian upper bounds on heat kernels of uniformly elliptic operators on bounded domains
The author obtains Gaussian upper bounds for heat kernels of higher order differential operators NEWLINE\[NEWLINEHf(x)=\sum_{|\alpha|\leq m,|\beta|\leq m}(-1)^{|\alpha|}D^\alpha (a_{\alpha,\beta}(x)D^\beta f(x))NEWLINE\]NEWLINE with Dirichlet boundary conditions on bounded domains in \(\mathbb R^N\). The bounds exhibit explicitly the nature of the spatial decay of the heat kernel close to the boundary as well as the long-time exponential decay implied by the spectral gap. The coefficients of the operator supposed to be only bounded and measurable.
scientific article; zbMATH DE number 6090234