The [formula omitted] theory of divergence form elliptic operators under the Dirichlet condition

Abstract

Let A be the 2 m th-order elliptic operator of divergence form with bounded measurable coefficients defined in a domain Ω of R n with smooth boundary. For 1 < p < ∞ we regard A as a bounded operator from the L p Sobolev space H 0 m , p ( Ω ) to H - m , p ( Ω ) . Trying to extend the result which has been obtained only in the case Ω = R n or in the case m = 1 to the general case, we succeed in constructing the resolvent ( A - λ ) - 1 and estimating its operator norm for some λ when the leading coefficients are uniformly continuous. Applying the result for L p resolvents, we show that the operator associated with A in L p ( Ω ) generates an analytic semigroup and obtain exponential decay estimates for the heat kernel, the resolvent kernel and their derivatives of order up to m - 1 . We also give a perturbation theorem for heat kernels.