Lecture 9

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Dear Participants,

in this lecture we complete the analysis of the parabolic equation D_tu-Au=f in the whole \mathbb R^d showing that a semigroup of bounded operators (T(t)) can be associated with this equation. We also prove some uniform estimates for the function T(t)f in spaces of Hoelder continuous functions, which are completely analogous to those proved in the case of the Gauss-Weierstrass semigroup. The proof is based on some results from the previous lecture and it is much more technical and let's say somehow more difficult than the one we provided for the Gauss-Weierstrass semigroup. But an explicit formula for the Gauss-Weierstrass semigroup is available and this makes things easier. On the contrary, an explicit formula for the semigroup (T(t)) considered in this lecture is not available in general...

As usually, we hope you enjoy the lecture :-).

Lecture9.pdf

We kindly ask to the team from Ljubljana to upload on the web page of the ISEM the official solutions of the exercises of this lecture. Kind regards. Your virtual lecturers

Abdelaziz and Luca