# Lecture 8

Dear Participants,

sorry for one day (and a few hours) delay in sending you the new lecture. In this lecture, we first prove the existence of classical solutions to the nonhomogeneous Cauchy problem (7.1) and Optimal Schauder estimates, using the continuation method writing a segment of elliptic operators joining the Laplacian to our (beloved) operator A. Next, in Section 8.2, we first convince you by an example that, differently from the 1D setting, in more dimensions, if we assume the data just bounded and continuous, in general we can not expect that the Cauchy problem (7.1) admits a classical solution. On the other hand, we can weaken a bit our assumptions on f and g getting a weak regularity result on the solution to the Cauchy problem (7.1), where now the boundedness of the time derivative and the spatial derivatives near t=0 is no more guaranteed. As a byproduct, we estimate in Theorem 8.2.3 the behaviour of several Hoelder norms of u as t tends to 0. Being the proof rather technical, we have decided to move to Appendix B the interior Schauder estimates, which are obtained starting from the Optimal Schauder estimates in the whole . Such estimates will play a crucial role in the study of the Cauchy problems associated with elliptic operators with unbounded coefficients. Now, it's time to stop writing and it's time to download the lecture.

Enjoy the reading!

This time we kindly ask the team from Karlsruhe to provide the official solutions of the exercises of the lecture.

Kind regards.

Abdelaziz & Luca