Other Problems with Linear Boundary Conditions in a Domain with a Small Hole

In Chapter 8 we introduced the Functional Analytic Approach. To do so, we considered its application to a Dirichlet problem for the Laplace equation in a bounded domain with a small hole that shrinks to an interior point. In this chapter we are going to see that the same method also works with other boundary value problems with linear boundary conditions, provided that we introduce a suitable integral representations of the solution. Instead, nonlinear problems, even with linear boundary conditions, require a specific modification of our approach. This is for example the case of the eigenvalue problems, which are not linear with respect to the pair eigenvalue-eigenfunction. To give some examples, we will consider a mixed problem for the Laplace equation, a mixed problem for the Poisson equation, and a Steklov eigenvalue problem. All of these problems will be defined in a perforated domain with a shrinking hole, so our geometric setting will still be that of Chapter 8 . We will see that the analysis of the mixed problem for the Laplace equation requires no particular modification of the method introduced for the Dirichlet problem. The mixed Poisson problem can then be studied using the results obtained for the mixed Laplace problem and for the volume potentials of Chapter 7 . To study the Steklov problem, on the other hand, we will have to do something new: we no longer can use Proposition A.15 on the analyticity of the inversion map and instead will resort to the Implicit Function Theorem A.19 for real analytic maps.