ABSTRACT
Simplifying, we could say that the goal of this chapter is the study of the Helmholtz equation Δ u + k 2 u = f , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/umath8_1.jpg"/> with different sets of boundary conditions. In reality, we are going to develop the use of the Fredholm theory for problems that can be understood as a compact perturbation of a coercive problem. This will allow us to consider low order terms as perturbations and derive a theory that deals with existence and uniqueness by looking at: (a) well-posedness of a problem without the perturbations; (b) uniqueness of solution. We will derive the Fredholm alternative and its applications to variational problems in two steps. In the first step, we will deal with self-adjoint problems, using a very simple proof of the Fredholm alternative that holds for self-adjoint problems. Examples will include the Helmholtz equation with Dirichlet or Neumann boundary conditions, and we will revisit Neumann and Robin boundary value problems for the Laplace equation, allowing for lack of uniqueness of solutions and deriving compatibility conditions on the data in a systematic way. In a second step, we will deal with more general operators and variational formulations, which will include the convection-diffusion equation and the Helmholtz equation with impedance boundary conditions.
