On L ^p-Theory of C ₀-Semigroups Generated by Elliptic Second Order Differential Expressions

Abstract

We study well-posedness in L ^p of the Cauchy problem for second order parabolic equations with time-independent measurable coefficients by means of constructing corresponding C ₀-semigroups. Lower order terms are considered as form-bounded perturbations of the generator of the symmetric Markov semi-group associated with the Dirichlet form. It is shown that the C ₀-semigroup corresponding to the Cauchy problem exists in a certain interval in the scale of L ^p-spaces which depends only on form-bounds of perturbations. Examples show that this interval cannot be extended. A new method is developed for studying the L ^p-independence of the spectrum. The L ^p-independence of the spectrum is proved for non-symmetric generators of semigroups, without assumptions of Gaussian-type estimates.

This note is a short survey of the results on L ^p-properties of C ₀-semigroups generated by second order elliptic operators with measurable coefficients. This theory is being extensively developed nowadays mainly because of applications to partial differential equations. We start with some properties of symmetric Markov semigroups and their generators for a variety of examples of which we refer to [2]. In particular, selfadjoint second order elliptic operators in divergence form are operators, one of possible application of which is perturbation by zero order terms (potentials). The last (and the main) section deals with second order operators containing first and zero order terms. We construct semigroups on L ^p-spaces and derive some of their important properties.