Subspace Interpolation with Applications to Elliptic Regularity

Abstract

In this paper, we prove new embedding results by means of subspace interpolation theory and apply them to establishing regularity estimates for the biharmonic Dirichlet problem and for the Stokes and the Navier–Stokes systems on polygonal domains. The main result of the paper gives a stability estimate for the biharmonic problem at the threshold index of smoothness. The classic regularity estimates for the biharmonic problem are deduced as a simple corollary of the main result. The subspace interpolation tools and techniques presented in this paper can be applied to establishing sharp regularity estimates for other elliptic boundary value problems on polygonal domains.

Keywords:

• Biharmonic operator
• Elliptic regularity
• Navier–Stokes systems
• Subspace interpolation
• Shift theorems
• Stokes systems
• Threshold index of smoothness

AMS Subject Classification:

• 35B30
• 35B65
• 35Q30
• 46B70

ACKNOWLEDGMENTS

Many thanks go to Jim Bramble, Bruce Kellogg, Joe Pasciak, and Jinchao Xu for their guidance and the valuable discussions on the subject.

This work was partially supported by NSF (DMS-0713125).