Lecture 11

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Dear Participants,

first of all our best wishes for a great 2017. We begin the new year by completing the analysis of Dirichlet-Cauchy problems in the halfspace. This lecture is rather technical, we apologize for that, but this effort will be paid back to you in the sense that we are able to improve the result in Theorem 10.2.2 by removing the extra condition on f and g on the boundary of \mathbb R^d_+, replacing it with the necessary (and sufficient) compatibility condition. Unfortunately, this requires an amount of additional work and a slight change on the strategy described in the last lecture to attack the problem. The compatibility conditions to be satisfied by data create some problems in adapting the same arguments as in Lecture 8 to prove apriori optimal Schauder estimates for the solution to the Cauchy problem (10.1). This problem is overcome by removing..... the problem ;-) i.e., by proving Schauder estimates for solutions to a boundary value problem set in the time domain (-\infty,T]. We anticipate you that we are modyfing the previous lectures according to your very valuable suggestions on the discussion board and also Lecture 10 will be modified according to what we wrote above. This aimed to make the lectures quite.... "perfect" (clearly, we are joking, since we are very far away from perfection ;-). Finally, we promise you to pay much more attention to the discussion board and we apologize for our lack at the end of last year. We also apologize for this long email message. ;-) So, it's time to download and read the lecture.

Lecture11.pdf

This time we kindly ask the team from Darmstadt to prepare the official solutions of the exercises in the lecture. Kind regards.

Your virtual lecturers.

Abdelaziz & Luca