# Lecture 6

Dear Participants,

this week we complete the analysis of the heat equation. Using the technical material in the previous lectures, we prove optimal Schauder estimates for the solution to the Cauchy problem $D_tu=\Delta u+g$ in $[0,T]\times\mathbb R^d$ such that $u(0,\cdot)=f$. Namely, we show that, if $g$ is $\alpha$-Hölder continuous with respect to the parabolic distance of $\mathbb R^{d+1}$ in $[0,T]\times\mathbb R^d$, i.e., with respect to the distance $d((t_2,x_2),(t_1,x_1))=\sqrt{|t_2-t_1|+|x_2-x_1|^2}$ and $f\in C^{2+\alpha}_b(\mathbb R^d)$ then the above Cauchy problem admits a (unique) solution $u\in C^{1,2}([0,T]\times\mathbb R^d)$ such that all its derivatives have the same degree of smoothness of $g$ (we simply say that $u\in C^{1+\alpha/2,2+\alpha}_b([0,T]\times\mathbb R^d)$).

Another equivalent way of stating this result is the following: if a function $u\in C^{1,2}([0,T]\times\mathbb R^d)$ is such that $D_tu-\Delta u\in C^{\alpha/2,\alpha}_b([0,T]\times\mathbb R^d)$ and $u(0,\cdot)\in C^{2+\alpha}_b(\mathbb R^d)$ then $u$ is smoother than $C^{1,2}([0,T]\times\mathbb R^d)$: it belongs to $C^{1+\alpha/2,2+\alpha}_b([0,T]\times\mathbb R^d)$.

Besides their own interest, from a mathematical view point, optimal Schauder estimates are also very important in the study of nonlinear equations. In this lecture, we see an example left as an exercise to you, but in one of the forthcoming lectures we will also see some other concrete example where optimal Schauder estimates will be used to solve nonlinear parabolic equations with nonlocal terms.

Now, it is time to choose the team which will be responsible for the official solutions to the exercises of this lecture, and our favorite team for this week is the team from Lecce

Enjoy the lecture and the ISEM