Abstract differential equations of elliptic type with general Robin boundary conditions in Hölder spaces

Abstract

In this article, we prove some new results on abstract second-order differential equations of elliptic type with general Robin boundary conditions. The study is performed in Hölder spaces and uses the well-known Da Prato–Grisvard sum theory. We give necessary and sufficient conditions on the data to obtain a unique strict solution satisfying the maximal regularity property. This work completes the ones studied by Favini et al. [A. Favini, R. Labbas, S. Maingot, H. Tanabe, and A. Yagi, Necessary and sufficient conditions in the study of maximal regularity of elliptic differential equations in Hölder spaces, Discrete Contin. Dyn. Syst. 22 (2008), pp. 973–987] and Cheggag et al. [M. Cheggag, A. Favini, R. Labbas, S. Maingot and A. Medeghri, Sturm-Liouville problems for an abstract differential equation of elliptic type in UMD spaces, Differ. Int. Eqns 21(9–10) (2008), pp. 981–1000].

Keywords:

• second-order abstract elliptic differential equations
• Robin boundary conditions
• analytic semigroup
• maximal regularity
• concrete transmission problems

AMS Subject Classifications::

• 35J20
• 35J40
• 47D06

Acknowledgements

The authors thank the referees for their useful comments and remarks.