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Maximal \(L_{p}\)-regularity of non-local boundary value problems

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Abstract

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.

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Notes

  1. For a definition of \(B\)-convexity we refer the reader to [16]. For us it will be sufficient to know that \(L_p\)-spaces with \(1<p<\infty \) are \(B\)-convex.

  2. For the definition of these properties we refer the reader to [17] or [5]. For us it is sufficient to know that scalar-valued \(L_p\)-spaces, \(1<p<\infty \), have these properties.

  3. This principle states that the inequality

    $$\begin{aligned} \sum _{z_1,\ldots ,z_N=\pm 1}\left\| \sum _{j=1}^N z_j\alpha _jx_j\right\| ^q\le 2^q \sum _{z_1,\ldots ,z_N=\pm 1}\left\| \sum _{j=1}^N z_j\beta _jx_j\right\| ^q\qquad \qquad \end{aligned}$$

    holds true whenever \(\alpha _j,\beta _j\in {\mathbb C}\) with \(|\alpha _j|\le |\beta _j|\) and \(x_1,\ldots ,x_N\in X\) with arbitrary \(N\).

  4. The operator class on \(M\) instead of the half-space is again obtained by using a covering by local coordinate systems and a subordinate partition of unity and taking into account the global smoothing remainders defined in Definition 3.

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Acknowledgments

We thank the anonymous referees for their useful comments and one of them for pointing out a gap in the argumentation of Sect. 5.2.

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  1. Universität Konstanz, Fachbereich Mathematik und Statistik, Fach 193, 78457 , Konstanz, Germany

    Robert Denk

  2. Dipartimento di Matematica, Università di Torino, V. Carlo Alberto 10, 10123 , Turin, Italy

    Jörg Seiler

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  1. Robert Denk
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Correspondence to Jörg Seiler.

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Communicated by J. Escher.

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Denk, R., Seiler, J. Maximal \(L_{p}\)-regularity of non-local boundary value problems. Monatsh Math 176, 53–80 (2015). https://doi.org/10.1007/s00605-014-0669-4

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