Neumann-transmission problems for pseudodifferential Brinkman operators on Lipschitz domains in compact Riemannian manifolds

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The aim of this paper is twofold. On the one handwe construct Neumann-transmission kernels forpseudodifferential Brinkman operators. They are used to provide simple representations of thesolution to some transmission problems for the pseudodifferential Brinkman operator.On the other hand, we showthe well-posedness of a Neumann-transmission problem for twopseudodifferential Brinkman operators on adjacent Lipschitz domainsin a compact Riemannian manifold, with boundary data in some $L^p$, Sobolev or Besov spaces. We rely on thelayer potential theory in order to obtain an explicit representation of the solution to this problem.Compactness and invertibility results of associated layer potentialoperators on $L^p$, Sobolev and Besov spaces are also presented.