Edge problems on configurations with model cones of different dimensions

Abstract

Elliptic equations on configurations $W = W_{1} \cup \dots \cup W_{N}$ with edge $Y$ and components $W_{j}$ of different dimension can be treated in the frame of pseudo-differential analysis on manifolds with geometric singularities, here edges. Starting from edge-degenerate operators on $W_{j}$, $j=1,\dots,N$, we construct an algebra with extra `transmission' conditions on $Y$ that satisfy an analogue of the Shapiro-Lopatinskij condition. Ellipticity refers to a two-component symbolic hierarchy with an interior and an edge part; the latter one is operator-valued, operating on the union of different dimensional model cones. We construct parametrices within our calculus, where exchange of information between the various components is encoded in Green and Mellin operators that are smoothing on $W \setminus Y$. Moreover, we obtain regularity of solutions in weighted edge spaces with asymptotics.