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Interface and mixed boundary value problems on n-dimensional polyhedral domains



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687 Documenta Math Interface and mixed boundary value problems on n dimensional polyhedral domains Constantin Ba cut a 1 Anna L Mazzucato2 Victor Nistor3 and Ludmil Zikatanov4 Received December 1 2008 Communicated by Eckhard Meinrenken Abstract Let Z be arbitrary We prove a well posedness result for mixed boundary value interface problems of second order positive strongly elliptic operators in weighted Sobolev spaces Ka on a bounded curvilinear polyhedral domain in a manifold M of dimension n The typical weight that we consider is the smoothed distance to the set of singular boundary points of Our model problem is P u div A u f in u 0 on D and D P u 0 on where the function A 0 is piece wise smooth on the polyhedral decomposition j j and D N is a decomposition of the boundary into polyhedral subsets corresponding respectively to Dirichlet and Neumann boundary conditions If there are no interfaces and no adjacent faces with Neumann boundary conditions our main result gives an isomorphism 1 1 for P Ka 1 u 0 on D D P u 0 on N Ka 1 0 and a for some 0 that depends on and P but not on If interfaces are present then we only obtain regularity on each subdomain j Unlike in the case of the usual Sobolev spaces can be arbitrarily large which is useful in certain applications An important step in our proof is a regularity result which holds for general strongly elliptic operators that are not necessarily positive The regularity result is based in turn on a study of the geometry of our polyhedral domain when endowed with the metric dx 2 where is the weight the smoothed distance to the singular set The wellposedness result applies to positive operators provided the interfaces are smooth and there are no adjacent faces with Neumann boundary conditions 1 The 2 The work of C Bacuta is partially supported by NSF DMS 0713125 work of A Mazzucato is partially supported by NSF grant DMS 0405803 and DMS 0708902 3 The work of V Nistor is partially supported by NSF grant DMS 0555831 DMS



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