in this lecture we start the analysis of parabolic equation with homogeneous Dirichlet boundary conditions in the "half-space". The analysis of these equations on the half-space and the previous lectures on parabolic equations on the whole space are the key tools to analyze Cauchy problems on general bounded (and smooth enough) domains. The lectures is concerned with the Laplacian which as you now may guess is the simplest situation and the starting point for the analysis of much more complicated Cauchy problems. Indeed, by a reflection the Cauchy problem on the half-space can be related to the Cauchy problem on the whole space and so we can apply the results in Lectures 4 to 6.
Since we realised that a huge amount of material was needed to discuss already, we decided to give you and ourselves a short break. We will meet on the 10th January.
We would like to convey you our warmest wishes for a marvellous end of 2016 and a still better 2017. Your virtual lecturers.
Abdelaziz and Luca