The Poisson equation

Question

I see the following theorem in Lihe Wang's A geometric approach to the Calderon--Zygmund estimates $$ \triangle u=f\quad in \quad \> B_2. \>\quad \quad \quad \quad (1) $$

Lemma 7: There is a constant $N_1$ so that for any $ε > 0$, $∃ δ = δ(ε) > 0$ and if u is a solution of (1) in a domain $\Omega \supset B_4$,with $$\{\mathcal{M}(|f|^2)\leq δ^2\}\cap \{\mathcal{M}|D^2u|^2\leq 1\}\cap B_1\neq \emptyset$$ then $$ |\{\mathcal{M}|D^2u|^2> N_1^2\}\cap B_1|< \varepsilon|B_1|$$

Proof From condition (19), we see that there is a point$ x_0\in B_1$ so that:

$$\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}|D^2u|^2 \leq 1\quad and \quad \frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}|f|^2 \leq \delta^2$$ for all $B_r(x_0) \subset \Omega$, and consequently we have $$\frac{1}{|B_4|}\int_{B_4}|D^2u|^2 \leq 2^n \quad and \quad \frac{1}{|B_4|}\int_{B_4}|f|^2 \leq 2^n \delta^2$$ Then $$\frac{1}{|B_4|}\int_{B_4}|\nabla u-\overline{\nabla u}_{B_4}|^2 \leq c_1 $$

Let $v$ be the solution of the following equation $$ \triangle v=0$$ $$v=u-(\overline{\nabla u})_{B_4} \cdot x-\overline{ u}_{B_4} \>on\> \partial B_4$$ Then by the minimality of harmonic function with respect to energy in $B_4$, $$\int_{B_4}|\nabla v|^2 \leq \int_{B_4}|\nabla u-\overline{\nabla u}_{B_4}|^2 \leq c_1$$

I would like to see a proof of: $$\int_{B_4}|\nabla v|^2 \leq \int_{B_4}|\nabla u-\overline{\nabla u}_{B_4}|^2 $$

Answer 1

By the Dirichlet principle harmonic functions in $W^{1,2}(B_4)$ minimize the Dirichlet energy $\int_{B_4}|\nabla u|^2$ among all function in $W^{1,2}(B_4)$ with the same boundary data. Since $v\in W^{1,2}(B_4)$ has the same boundary data as $u-(\overline{\nabla u})_{B_4} \cdot x-\overline{ u}_{B_4}$ we have that $$ \int_{B_4}|\nabla v|^2 \leq \int_{B_4} |\nabla(u-(\overline{\nabla u})_{B_4} \cdot x-\overline{ u}_{B_4})|^2 =\int_{B_4}|\nabla u-\overline{\nabla u}_{B_4}|^2 \leq c_1. $$