Lecture 14

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Dear Participants,

this week, we consider more general elliptic operators {\mathcal A} with unbounded coefficients on \mathbb R^d. Under very reasonable assumptions, we show that the homogeneous Cauchy problem associated to such operators admits a classical solution, which is bounded in each strip [0,T]\times\mathbb R^d. Differently from the case of bounded coefficients, in general we can not expect uniqueness of the classical solution which is bounded in the strips [0,T]\times\mathbb R^d. In fact, in the one-dimensional setting we provide necessary and sufficient conditions for the above Cauchy problem to be uniquely solvable. It turns out that if {\mathcal A}u=u''-x^3u' then, uniqueness holds, whereas if {\mathcal A}u=u''+x^3u' then, for any f\in C_b(\mathbb R) we can find infinitely many bounded classical solution to the equation D_tu={\mathcal A}u which satisfy the condition u(0,\cdot)=f!!! This a a typical feature of Cauchy problems associated with elliptic operators with unbounded coefficients. Under a stronger assumption on the coefficients of the elliptic operator (the existence of the so-called Lyapunov function) a generalized version of the maximum principle can be proved, and uniqueness follows!!! Here, the lecture to download


and do not forget to download also Appendix-B which has been updated with new results which we need in this lecture.

This week we kindly ask the team from Dresden to prepare the official solutions to the exercises of the lecture.

We inform all of you that this is the last but one lecture of this ISEM. Next, week we will conclude our analysis with........................................... Please, be patient!!! ;-)

Kind regards.

Your virtual lecturers

Abdelaziz & Luca