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: $$\int_{B_4}|\nabla v|^2 \leq \int_{B_4}|\nabla u-\overline{\nabla u}_{B_4}|^2 $$

  • $\begingroup$ This is basically Dirichlet's principle. If you look in Evans' PDE book, this is Theorem 17 in chapter 2, section 2.2.5b. If two functions have the same Dirichlet data on the boundary of a set, then the one that is harmonic has smaller $\dot{H}^1$ energy. $\endgroup$ – Willie Wong Oct 1 '18 at 15:52
  • $\begingroup$ @WillieWong in Theorem 17 (in chapter 2) the function $u\in C (\overline{U})$, is not for a function $u\in W^{1,2}(U)$ ,while $v$ can be does not belong to $ C (\overline{U})$ $\endgroup$ – deval sidi Dec 5 '18 at 18:22
  • $\begingroup$ as Piotr notes in his answer, Dirichlet's principle extends also for $W^{1,2}$ functions. See, e.g., these lecture notes. $\endgroup$ – Willie Wong Dec 5 '18 at 21:05
  • $\begingroup$ @WillieWong my question(see Theorem 1) Is: if we have $\omega \in L^{2}(\Omega)$ then there is $u$ a function such as $\omega=\nabla u$ in the sense of distributions. $\endgroup$ – deval sidi Mar 20 '19 at 21:40

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 $$ \int_{B_4}|\nabla v|^2 \leq \int_{B_4} |\nabla(u-(\overline{\nabla u})_{B_4} \cdot x-\overline{ u}_{B_4})|^2 =\int_{B_4}|\nabla u-\overline{\nabla u}_{B_4}|^2 \leq c_1. $$

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