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We propose an isogeometric mortar method to fourth order elliptic problems.In particular we are interested in the discretization of the biharmonicequation on $C^0$-conforming multi-patch domains and we exploit the mortartechnique to weakly enforce $C^1$-continuity across interfaces. In order toobtain discrete inf-sup stability, a particular choice for the Lagrangemultiplier space is needed. Actually, we use as multipliers splines of degreereduced by two, w.r.t. the primal spline space, and with merged elements in theneighbourhood of the corners. In this framework, we are able to show optimal apriori error estimates. We also perform numerical tests that reflecttheoretical results.

Isogeometric multi-patch $C^1$-mortar coupling for the biharmonic equation