On the numerical range of second-order elliptic operators with mixed boundary conditions in $L^p$

We consider second-order elliptic operators with real, nonsymmetric coefficient functions which are subject to mixed boundary conditions. The aim of this paper is to provide uniform resolvent estimates for the realizations of these operators on $L^p$ in a most direct way and under minimal regularity assumptions on the domain. This is analogous to the main result in Chill et al. (C R Acad Sci Paris 342:909–914, 2006). Ultracontractivity of the associated semigroups is also considered. All results are for two different form domains realizing mixed boundary conditions. We further consider the case of Robin instead of classical Neumann boundary conditions and also allow for operators inducing dynamic boundary conditions. The results are complemented by an intrinsic characterisation of elements of the form domains inducing mixed boundary conditions