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We introduce an $R$-sectoriality perturbation technique for non-commutingoperators defined in Bochner spaces. Based on this and on bounded$H^{\infty}$-functional calculus results for the Laplacian on manifolds withconical singularities, we show maximal $L^{q}$-regularity for the Laplacian onmanifolds with edge type singularities in appropriate weighted Sobolev spaces.As an application, we consider the porous medium equation on manifolds withedges and show short time existence, uniqueness and maximal regularity for thesolution. We also provide space asymptotics near the singularities in terms ofthe local geometry.

Maximal $L^{q}$-regularity for the Laplacian on manifolds with edges