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 \(\mathrm {L}^p(\Omega )\). Additional properties like analyticity of the semigroup, \(\mathrm {H}^\infty \)-calculus and maximal regularity are also 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.

给定一个复杂的椭圆系数函数，我们研究对应的二阶散度形式算子的p值，辅以狄利克雷、诺依曼或混合边界条件，在\(\mathrm {L}^p(\欧米茄 )\)。还讨论了其他性质，如半群的解析性、\(\mathrm {H}^\infty \) -微积分和最大正则性。最后，我们证明了实系数的扰动结果，该结果给出了系数的小虚部的p的整个范围。我们的结果基于最近的p椭圆度概念、反向 Hölder 不等式和实系数的高斯估计。