Documenta Math. 687Interface and mixed boundary value problemson n-dimensional polyhedral domainsConstantin B˘acut¸˘a1, Anna L. Mazzucato2,Victor Nistor3, and Ludmil Zikatanov4Received: December 1, 2008Communicated by Eckhard MeinrenkenAbstract. Let µ ∈ Z+be arbitrary. We prove a well-posednessresult for mixed boundar y value/interface problems of second-order,positive, strongly elliptic operators in weighted Sobolev spa ces Kµa(Ω)on a bounded, curvilinear po lyhedral domain Ω in a manifold M ofdimension n. The typical weight η that we consider is the (smoothed)distance to the set of s ingular boundary points of ∂Ω. Our modelproblem is P u := − div(A∇u) = f, in Ω, u = 0 on ∂DΩ, andDPνu = 0 on ∂νΩ, where the function A ≥ ǫ > 0 is piece-wise smoothon the polyhedral decompo sition¯Ω = ∪j¯Ωj, and ∂Ω = ∂DΩ ∪ ∂NΩis a decomposition of the boundary into polyhedral subsets corre-sponding, respectively, to Dirichlet and Neumann boundary condi-tions. If there are no interfaces and no adjacent faces with Neu-mann boundary conditions, our main result gives an isomorphismP : Kµ+1a+1(Ω) ∩ {u = 0 on ∂DΩ, DPνu = 0 on ∂NΩ} → Kµ−1a−1(Ω) forµ ≥ 0 and |a| < η, for some η > 0 that depe nds on Ω and P butnot on µ. I f interfaces are pr esent, then we only obtain regularity oneach subdomain Ωj. Unlike in the case of the usual Sobolev spaces,µ can be arbitrarily large, which is useful in certain applications. Animportant step in our proof is a regu larity result, which holds for gen-eral strongly elliptic operators that are not necessarily positive. Theregular ity result is based, in turn, on a study of the geometry of ourpolyhedral domain when endowed with the metric (dx/η)2, where ηis the weight (the smoothed distance to the singular set). The well-posedness result applies to positive operators, provided the interfacesare smooth and there are no adjacent faces with Neumann boundaryconditions.1The work of C. Bacuta is partially supported by NSF DMS-0713125.2The work of A. Mazzucato is partially supported by NSF grant DMS-0405803 and DMS-0708902.3The work of V. Nistor is partially supported by NSF grant DMS-0555831, DMS-0713743,and OCI-0749202.4The work of L. Zikatanov is partially supported by NSF grant DMS-0810982 and OCI-0749202.Documenta Mathematica 15 (2010) 687–745688 C. Bacuta, A. L. Mazzucato, V. Nistor, L. Zikatanov2010 Mathematics Subject Classification: Primary 3 5J25; Secondar y58J32, 52B70 , 51B25.Keywords and Phrases: Polyhedral domain, elliptic equations,mixed boundary conditions, interface, weighted Sobolev spaces, well-posedness, L ie manifold.IntroductionLet Ω ⊂ Rnbe an open, bounded set. Consider the boundary value problem(∆u = f in Ωu|∂Ω= g, on Ω,(1)where ∆ is the L aplace operator. For Ω smooth, this boundary va lue problemhas a unique solution u ∈ Hs+2(Ω) depending continuously on f ∈ Hs(Ω) andg ∈ Hs+3/2(∂Ω), s ≥ 0. See the books of E vans [25], L ions and Magenes [49],or Taylor [72] for proofs of this basic and well known result.It is also well known that this result does not extend to no n-smooth domainsΩ. For instance, Jerison and Kenig prove in [35] that if g = 0 and Ω ⊂ Rn,n ≥ 3, is an open, bounded set such that ∂Ω is Lipschitz, then Equation (1) hasa unique solution in Ws,p(Ω) depending continuously on f ∈ Ws−2,p(Ω) if, andonly if, (1/p, s) belongs to a certain explicit hexagon. They also prove a similarresult if Ω ⊂ R2. A consequence of this result is that the smoothness of thesolution u (measured by the order s of the Sobolev space Ws,p(Ω) containingit) will no t exceed, in general, a certa in bound that depends on the domain Ωand p, even if f is smo oth.In addition to the Jerison and Kenig paper mentioned above, a deep analysis ofthe difficulties that arise for ∂Ω Lips chitz is contained in the papers o f Babuˇska[4], Baouendi and Sj¨ostrand [9], B˘acut¸˘a, Bramble, and Xu [14], Babuˇska andGuo [31, 30], Brown and Ott [13], Jerison and Kenig [33, 34], Kenig [38],Kenig and Toro [39], Koskela, Koskela and Zhong [43, 44], Mitrea and Taylor[58, 60, 61], Verchota [73], and others (see the references in the aforementionedpapers). Results more specific to curvilinear polyhedral domains a re c ontainedin the pa pers of Costabel [17], Dauge [19], Elschner [20, 21], Kondratiev [41, 42],Mazya and Rossmann [54], Rossmann [63] and others. Excellent references arealso the monog raphs of Grisvard [27, 28] as well as the recent books [45, 46,52, 53, 6 2], where more references can be found.In this paper, we consider the boundary value problem (1) when Ω is a boundedcurvilinear polyhedral domain in Rn, or, more generally, in a manifold M of di-mension n and, Poisson’s equation ∆u = f is replaced by a positive, stronglyelliptic scalar equation. We define curvilinear polyhedral domains inductivelyin Section 2. We allow p olyhedral domains to be disconnected for technical rea-sons, more precis e ly, for the purpose of defining them inductively. Our results,however, are formulated for connected polyhedral domains. Many polyhedralDocumenta Mathematica 15 (2010) 687–745Boundary value problems 689domains are Lips chitz domains, but not all. This fact is discussed in detail byVogel and Verchota in [74], where they also prove that the harmonic measure isabsolutely continuous with respect to the Lebesgue measure on the boundaryas well as the solvability of Equation (1) if f = 0 and g ∈ L2−ǫ(∂Ω), thusgeneralizing several earlier, classical results. See also the excellent book [50].The generalized poly he dra we considered are of combinatorial type if no cracksare present. (For a discussion of more general domains, see the references[68, 74, 7 5].)Instead of working with the usual Sobolev spaces, as in several of the papersmentio ne d above, we shall work in some weighted analogues of these pape rs.Let Ω(n−2)⊂ ∂Ω be the set of singular (or non-smooth) boundary points of Ω,that is, the s et of points p ∈ ∂Ω such ∂Ω is not smooth in a neighborhood of p.We shall denote by ηn−2(x) the distance from a point x ∈ Ω to the set Ω(n−2).We agree to take ηn−2= 1 if there are no such points, that is, if ∂Ω is smooth.We then consider the weighted Sobolev spacesKµa(Ω) = {u ∈ L2loc(Ω), η|α|−an−2∂αu ∈ L2(Ω), for all |α| ≤ µ}, µ ∈ Z+, (2)which we endow with the induced