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

Let $\mu \in \ZZ_+$ 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 $\Kond{\mu}a(\Omega)$ on
a bounded, curvilinear polyhedral domain $\Omega$ in a manifold $M$ of
dimension $n$. The typical weight $\eta$ that we consider is the (smoothed)
distance to the set of singular boundary points of $\pa \Omega$. Our model
problem is $Pu:= - \dive(A \nabla u) = f$, in $\Omega$, $u = 0$ on $\pa_D
\Omega$, and $D^P_\nu u = 0$ on $\pa_\nu \Omega$, where the function $A
\ge \epsilon > 0$ is piece-wise smooth on the polyhedral decomposition
$\Bar\Omega = \cup_j \Bar\Omega_j$, and $\pa \Omega = \pa_D \Omega \cup
\pa_N \Omega$ 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 $P : \Kond{\mu+1}{a+1}(\Omega)
\cap {u=0 on \pa_D \Omega, \ D_\nu^P u=0 on \pa_N \Omega}
\to \Kond{\mu-1}{a-1}(\Omega)$ for $\mu \ge 0$ and $|a|<\eta$, for some
$\eta>0$ that depends on $\Omega$ and $P$ but not on $\mu$. If interfaces
are present, then we only obtain regularity on each subdomain $\Omega_j$.
Unlike in the case of the usual Sobolev spaces, $\mu$ 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 $(dx/\eta)^2$, where $\eta$ is the weight (the smoothed
distance to the singular set). The well-posedness result applies to positive
operators, provided the interfaces are smooth and there are no adjacent
faces with Neumann boundary conditions.

2010 Mathematics Subject Classification: Primary 35J25; Secondary 58J32, 52B70, 51B25.

Keywords and Phrases: Polyhedral domain, elliptic equations, mixed boundary conditions, interface, weighted Sobolev spaces, well-posedness, Lie manifold.

Full text: dvi.gz 115 k, dvi 276 k, ps.gz 946 k, pdf 518 k.

Home Page of DOCUMENTA MATHEMATICA