Lecture 7

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Dear Participants,

we are back to you with the 7th lecture. This week we deal with more general elliptic operators A on \mathbb R^d. Our aim consists in extending the results we proved in the previous lectures for the Laplacian to the operator A. Two are the main tools to achieve this goal: apriori estimates for solutions to the parabolic equation D_tu=Au+g in [0,T]\times \mathbb R^d and a powerful argument from functional analysis, the so-called continuity method, which, roughly speaking says the following: if we have a segment of bounded linear operators L_t (t\in [0,1]) from a Banach space X into a Banach space Y, which satisfy a suitable estimate, with a constant, independent of t, and the first operator of the segment (say L_0) is invertible, then all the operators L_t are invertible. The a priori estimates are obtained in some steps. First, by easy arguments, they are proved in the case when A has only second-order terms and the diffusion coefficients are constant. Based on this result, the apriori estimates are then proved in the general case by "freezing the coefficients". This part is rather technical but very interesting since most of the interpolation results, which we proved in the previous lectures, are used. We hope you enjoy this lecture!!!

This time we ask the team Marrakesh to prepare the official solutions to the exercises of Lecture 7.

Kind regards.

Abdelaziz and Luca