Abstract
Uniform a priori estimates for parameter-elliptic boundary value problems are well-known if the underlying basic space equals \(L^{p}(\Omega )\). However, much less is known for the \(W_{p}^{s}(\Omega )\)-realization, s > 0, of a parameter-elliptic boundary value problem. We discuss a priori estimates and the generation of analytic semigroups for these realizations in various cases. The Banach scale method can be applied for homogeneous boundary conditions if the right-hand side satisfies certain compatibility conditions, while for the general case parameter-dependent norms are used. In particular, we obtain a resolvent estimate for the general situation where no analytic semigroup is generated.
Dedicated to Yoshihiro Shibata on occasion of his 60 th birthday
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Denk, R., Seger, T. (2016). Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity. In: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (eds) Recent Developments of Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0939-9_9
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