ABSTRACT
In Chapter 2 we explored the Dirichlet problem with homogeneous boundary conditions. It is natural to ask what happens if we want to apply boundary conditions which are not homogeneous. The situation is much more complicated, and we devote this chapter to answering that question. We thus want to explore the solvability theory for a problem of the form − Δ u = f in Ω , u = g on ∂ Ω . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/math4_1.jpg"/> Given the fact that we will be working in the Sobolev space H 1(Ω), this will require us to give a precise meaning to the concept of restriction to the boundary (we will call it a ‘trace’ operator). The construction is not entirely obvious and requires some prior technical work, which will help us better understand the Sobolev spaces H 1(Ω), under some new constraints on what the open set Ω can be. Here is the plan for this chapter:
We will relate the problems of the extension of H 1(Ω) functions to H 1 ( R d ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math4_5.jpg"/> functions with the possibility of having a dense subset of H 1(Ω) comprised of smooth functions.
Once the density results have been made clear, we will be ready to go to the boundary and take the trace of an H 1(Ω) function on the boundary of Ω. This will be a good moment to discuss some simple functional analytic tools about image norms.
We will next identify the kernel of the trace operator with the space H 0 1 ( Ω ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429507069/af768ad5-21aa-4c7b-88c6-5318b65e2dde/content/inline-math4_9.jpg"/> , thus proving that the homogeneous Dirichlet problem is a particular case of the nonhomogeneous problem (this might look like a trivial statement, but it is not) when the trace operator is defined.
66Finally, when all the tools are ready, we will be able to describe (4.1) in a rigorous way and prove its unique solvability and well-posedness.
