Gaussian upper bounds for heat kernels of second-order elliptic operators with complex coefficients on arbitrary domains

摘要

We consider second-order elliptic operators of the type A = - Sigma(k,j) D-j(a(kj)D(k)) + Sigma(k) b(k)D(k) - D-k(c(k) (.)) + a(0) acting on L-2(Omega) (Omega is a domain of R-d, d greater than or equal to 1) and subject to various boundary conditions. We allow the coefficients a(kj), b(k), c(k) and a(0) to be complex-valued bounded measurable functions. Under a suitable condition on the imaginary parts of the principal coefficients a(kj), we prove that for a wide class of boundary conditions, the semigroup (e(-tA))(tgreater than or equal to0) is quasi-L-p-contractive (1 < p < infinity). We show a pointwise dornination of (e(-tA))(tgreater than or equal to0) by a semigroup generated by an operator with real-valued coefficients and prove a Gaussian upper bound for the associated heat kernel.