| Abstract: |
| We extend the concept of p-ellipticity for scalar elliptic operators---introduced by us in 2016---to the case of elliptic systems with complex coefficients. Although our condition turns out to be equivalent to a condition for systems (equally named) by Dindos, Li and Pipher (2021), the two approaches are different and yield different results.
Our primary goal is to develop an algebraic and geometric framework for studying p-ellipticity in the systems setting. We investigate how the structural properties of fourth-order coefficient tensors determine the range of p-ellipticity, and how this range encodes the anisotropic and asymmetric features of the tensor. We also aim to explicitly determine or estimate the range of p-ellipticity for significant concrete systems and notable classes of tensors.
As an application of our theory, we establish contractivity on L^p of the semigroup generated by the associated elliptic operators, under Dirichlet, Neumann, and mixed boundary conditions. |
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