Maximal $L_p$-regularity of non-local boundary value problems

TL;DR

This paper establishes the maximal $L_p$-regularity for non-local boundary value problems by proving $ $-boundedness of certain operator families, with applications to the reduced Stokes equation in waveguides.

Contribution

It demonstrates $ $-boundedness of Green operators in the Boutet de Monvel calculus, leading to new maximal $L_p$-regularity results for non-local boundary problems.

Findings

$ $-boundedness of weakly and strongly parameter-dependent Green operators.

Maximal $L_p$-regularity for non-local boundary value problems.

Application to the reduced Stokes equation in waveguides.

Abstract

We investigate the $\mathcal R$-boundedness of operator families belonging to the Boutet de Monvel calculus. In particular, we show that weakly and strongly parameter-dependent Green operators of nonpositive order are $\mathcal R$-bounded. Such operators appear as resolvents of non-local (pseudodifferential) boundary value problems. As a consequence, we obtain maximal $L_p$-regularity for such boundary value problems. An example is given by the reduced Stokes equation in waveguides.