**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).

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