Explicit and uniform estimates for second order divergence operators on LP spaces

It is the aim of the talk to give – aside the Beurling/Deny approach – a consistent definition of second order divergence operators on spaces, even if the underlying domain is highly non-smooth, the boundary conditions are mixed and the coefficient function is real, bounded and elliptic – but not necessarily symmetric. In order to do this, one first proves that, under minimal assumptions, the resolvent transports the spaces with sufficiently large into. This shows that, for these, the part of the operator in possesses a domain which embeds into. Having this at hand, one can modify ideas of Cialdea/Maz’ya to include the numerical range in a certain sector. This leads to suitable resolvent estimates. Moreover, we prove that the resulting semigroup is contractive and analytic with explicitly determined holomorphy angle. Finally, a holomorphic calculus is established with (half) angle smaller than. This gives even maximal parabolic regularity via the Dore/Venni theorem.