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We investigate the $$\fancyscript{R}$$ -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 $$\fancyscript{R}$$ -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. [ABSTRACT FROM AUTHOR]

We study maximal L-regularity for a class of pseudodifferential mixed-order systems on a space-time cylinder $${\mathbb{R}^n \times \mathbb{R}}$$ or $${X \times \mathbb{R}}$$ , where X is a closed smooth manifold. To this end, we construct a calculus of Volterra pseudodifferential operators and characterize the parabolicity of a system by the invertibility of certain associated symbols. A parabolic system is shown to induce isomorphisms between suitable L-Sobolev spaces of Bessel potential or Besov type. If the cross section of the space-time cylinder is compact, the inverse of a parabolic system belongs to the calculus again. As applications, we discuss time-dependent Douglis-Nirenberg systems and a linear system arising in the study of the Stefan problem with Gibbs-Thomson correction. [ABSTRACT FROM AUTHOR]

In this note we show that the periodic b-equation can only be realized as a Euler equation on the Lie group Diff∞(S1) of all smooth and orientation preserving diffeomorphisms on the circle if b=2, i.e., for the Camassa–Holm equation. In this case the inertia operator generating the metric on Diff∞(S1) is given by A=1-∂x2. In contrast, the Degasperis–Procesi equation, for which b=3, is not a Euler equation on Diff∞(S1) for any inertia operator. Our result generalizes a recent result of Kolev [“Some geometric investigations on the Degasperis-Procesi shallow water equation,” Wave Motion 46, 412–419 (2009)]. [ABSTRACT FROM AUTHOR]

Given a manifold B with conical singularities, we consider the cone algebra with discrete asymptotics, introduced by Schulze, on a suitable scale of L p-Sobolev spaces. Ellipticity is proven to be equivalent to the Fredholm property in these spaces; it turns out to be independent of the choice of p. We then show that the cone algebra is closed under inversion: whenever an operator is invertible between the associated Sobolev spaces, its inverse belongs to the calculus. We use these results to analyze the behaviour of these operators on L p (B). [ABSTRACT FROM AUTHOR]