Elliptic interface problem without conditions on the interface

Question

Consider an open domain $U$ split in two non-overlapping subdomains: $U = U_1 \cup U_2$.

For a model case, consider a ball split in a smaller ball and an anulus.

Consider the following elliptic problem:

\begin{align*} -&\nabla ( A_1(x)\nabla u ) =f_1 \ &\text{ in } U_1\\ -&\nabla ( A_2(x)\nabla u )=f_2 & \text{ in } U_2\\ & u=g & \text{ on } \partial U \end{align*}

---

In the previous questions

• Elliptic problem on a domain split in two subdomains
• Boundary condition for elliptic problems and domain decomposition

a related problem was considered in the case of a prescribed jump or Neumann condition at the interface between $U_1$ and $U_2$.

---

In this question I wander about the general case without prescribed condition on $u$ at the interface.

• What references deal with such problems?
• What are the techniques to obtain existence and uniqueness results in this case (in the weak sense)?
• Indeed can we even get uniqueness without a condition at the interface? Why or why not?
• What are the minimal assumptions on $f_1$, $f_2$ and the domain that make the problem wellposed?

Answer 1

There are several aspects of your question.

In scientific computing, this amounts to "domain decomposition methods". Also "finite element methods" use such an idea, which is based on Ritz's method/Galerkin's method for a variational or weak formulation. Poincare's "balayage" also uses such decompositions. Sometimes one finds the terminology "interface problems" or the like. It's a class of standard problems in engineering.

In pure mathematics, such techniques are standard both in Harmonic Analysis/Potential Theory, and in the theory of elliptic partial differential equations. Davies in his book "Spectral Theory and Differential Operators", Cambridge Univ Press, uses e.g. minimax-methods on several domains.

A full treatment of elliptic operators, boundary conditions, boundary potentials, and resolvents is provided by the "Boutet-de-Monvel calculus". A monograph is e.g. Grubb: "Pseudodifferential Boundary Problems". You might also google for "transmission problems" and there is also a certain required "transmission condition" for regularity. Authors such as Bert-Wolfgang Schulze or Elmar Schrohe have studied such problems, also on domains with singularities. Another approach is by Richard B. Melrose, he had some lecture notes on his MIT-homepage about manifolds with corners. (Integral operators mapping functions on a manifold with boundaries to functions on a manifold with boundaries will have kernels defined on a manifold with singularities, so it is natural to consider manifolds with singularities right from the beginning.)

The general idea is to extend classical "Fredholm methods" by constructing parametrices and reducing the problem to compact perturbations of the identity operator, or at least to "regular" (e.g. smoothing) perturbations of the identity operator. To achieve this, various calculi are developped which should include the elliptic differential operators one is dealing with and their inverses or parametrices.

$L^2$ spaces of a disjoint union of two open sets can be rewritten as a direct sum of the $L^2$ spaces on each set. Zero Cauchy data as boundary conditions on the interface provide a kind of minimal operator, whose extensions might be sought. If these extensions should be selfadjoint differential operators, then there is a theory of selfadjoint extensions parametrized by boundary operators (-> "boundary triples"), see e.g. the book of Birman and Solomyak: "Spectral theory of selfadjoint operators in Hilbert space."

The literature is huge indeed, my answer is meant to give you names of some authors and some terminology to look up as a starting point.