WIAS Preprint No. 2590, (2019)
On the $L^p$-theory for second-order elliptic operators in divergence form with complex coefficients
- ter Elst, A. F. M.
- Haller-Dintelmann, Robert
- Rehberg, Joachim
- Tolksdorf, Patrick
2010 Mathematics Subject Classification
- 35J15 47D06 47B44
- Divergence form operators on open sets, p-ellipticity, sectorial, operators, analytic semigroups, maximal regularity, reverse Hölder inequalities, Gaussian estimates, De Giorgi estimates
Given a complex, elliptic coefficient function we investigate for which values of p the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on Lp(Ω). Additional properties like analyticity of the semigroup, H∞-calculus and maximal regularity arealso discussed. Finally we prove a perturbation result for real coefficients that gives the whole range of p's for small imaginary parts of the coefficients. Our results are based on the recent notion of p-ellipticity, reverse Hölder inequalities and Gaussian estimates for the real coefficients.
- J. Evol. Equ., 21 (2021), pp. 3963--4003, DOI 10.1007/s00028-021-00711-4 .