Markus Biegert, Michael Einemann and Markus Kunze

Regular form perturbations

Pages: 231-261
Received: 16 April 2009   
Accepted in revised form: 19 October 2009
Mathematics Subject Classification (2000): 35A15, 47A07, 47A55.

Keywords: Sectorial forms, perturbation, Kato-class.

Abstract: We present abstract results about the space regularity of solutions to elliptic and parabolic equations on L^p-spaces which are associated to perturbed sectorial forms a + b. As applications of our results, we introduce deGiorgi-Nash forms, which define quite general second order elliptic operators in divergence form. We give a wide class of examples of perturbations of such forms, such that the solutions of elliptic and parabolic equations associated to the perturbed operator are continuous. Furthermore, we prove that given any open subset Ω of R^N and any deGiorgi-Nash form a with principal coefficients in W^1,∞, there exists a potential V ∈ L^∞_{\mathrm{loc}} such that the operator associated to a + V generates a strongly continuous semigroup on Co(Ω).