Analysis and PDE : Caccioppoli, Morrey, Holder and ergodic of heat equation

Since I will study my PhD in SPDE, I have to find back my once very solid capacity in analysis and PDE. Maybe, the best way is to record some nice estimation that I have met in exercises and articles.

Functional inequality Morrey

We start from some functional inequality, the most classical but powerful tools of all analysis. We state the Morrey inequality. Generally speaking, if in domain $\Omega$ and a $L^2(\Omega)$ function, all the oscillation satisfies $\forall x \in \Omega, B_r(x) \subset \Omega,$
$$
 \frac{1}{|B_r(x)|} \int_{B_r(x)} |f(y) - (f)_{ B_r(x) } |^2 dy \leq M^2 r^{2 \alpha} 
$$
then we could say that this function has a a.e modification of class $C^{0, \alpha}$ Holder. A very easy corollary is to replace the oscillation by a gradient function and Poincare. The idea of proof comes from the Lebesgue differential theorem :
$$
a.e \lim_{r \rightarrow 0} (f)_{ B_r(x) } = f(x)
$$
so it suffices to pass all the estimation to the function $(f)_{ B_r(x) }$ and use "three difference trick" to the theorem.

One Holder interpolation

As the Soblev injection tells us, roughly speaking, the Holder space is a little better than all the $L^{p}$ space. Inspired by the interpolation theorem, we would like to obtain the Holder interpolation. One is
$$
\|f\|_{L^{\infty}(B_r)} \leq C \|f\|^{\frac{2\alpha}{d+2\alpha}}_{\underline{L}^{2}(B_r)}\left(r^{\alpha}[f]_{C^{0, \alpha}(B_r)}\right)^{\frac{d}{d+2\alpha}}
$$
This tells us that if we have the function Holder + $L^2$ implies also $L^{\infty}$. This may be seen as one part of the Soblev injection, but we recall a little its proof. Idea isn't difficult but wise : we find a small ball such that the value in it is at least 1/2 maximum and then we compare its $L^2$ norm. Thanks to the regularity, the radius of ball should not be so small and we get the result.

Caccioppoli inequality

Then we come to elliptic equation :
$$- \nabla \cdot (a(x) \nabla u(x)) = h$$
The regularity is one heart question in the research and one estimation used many many times in it is the Caccioppoli inequality
$$\|\nabla u\|_{\underline{L}^{2}(B_{r/2})} \leq C \left( \frac{1}{r} \|u -  (u)_{ B_r(x) }\|_{\underline{L}^{2}(B_r)} + \|h\|_{\underline{H}^{-1}(B_r)}\right)$$
The interpretation is very natural : this bound doesn't require the regularity of the coefficients but enlarge a little the domain. In fact, it tells us the interior of the solution is more regular while the outside may be a little pike.

Functional inequality Nash

Finally, we come to study the behavior of heat equation. We know that it will decrease generally, and one useful functional inequality is Nash that $\forall f \in L^1(\mathbb{R}^d) \bigcap H^1(\mathbb{R}^d)$ we have
$$
\|f\|^{1+2/d}_{L^2} \leq \|f\|^{2/d}_{L^1} \|\nabla f\|_{L^2}
$$For the solution of heat equation, we can apply easily the bound of $L^1 \rightarrow L^\infty, L^1 \rightarrow L^1$. By the decreasing of heat flow
$$
\frac{d}{dt}\|u(t, \cdot)\|^2_{L^2} = - \|\nabla u(t, \cdot)\|^2_{L^2}
$$
and Nash inequality we obtain
$$\|u(t, \cdot)\|_{L^2} \leq C t^{-d/4}\|u(0, \cdot)\|_{L^1}$$
The interpolation works and we obtain
$$\|u(t, \cdot)\|_{L^2} \leq C t^{-\frac{d}{2}(1-1/p)}\|u(0, \cdot)\|_{L^1}$$

Ergodic of heat equation

Finally, we would like to study the ergodic property. We suppose that the initial data $u(0, \cdot)$ is Z-periodic. In this case, we have $\forall t > 1$
$$
\|u(t, \cdot) - (u)_{\Box}\|_{L^{\infty}(\Box)} \leq \exp(-ct)
$$
Idea comes from the $L^2$ estimates and Holder estimates. The former is the result of decreasing of heat flow and the latter is the result of convolution. Then the two combine. We remark that when the time is small, we could not expect a better one since the average is of range $1/\sqrt{t}$ and could not regularize better.