Application of Fubini's theorem (in a proof of energy minimizing harmonic maps)

Question

Let $u\in H^1(B_1,S^k)$, where $B_1$ is the open unit ball in $\mathbb{R}^n$ and $S^k$ is the unit sphere in $\mathbb{R}^{k+1}$. Suppose that $u$ is a minimizer for the Dirichlet energy functional $$ I[w]=\int_{B_1}|Dw|^2dx $$ I have to prove that I can find some $r\in (0,1/2)$ such that $$ \int_{\partial B_r}|u-\bar{u}|^2\, \mathrm{d}x\le 3\int_{B_1} |u-\bar{u}|^2\, \mathrm{d}x $$ where $$ \bar{u}=\frac{1}{|B_1|}\int_{B_1}u\,dx $$

Answer 1

Suppose the claim is not true. Then for all $r\in(0,1/2)$ you have $$ \int_{\partial B_r}|u-\bar{u}|^2\, \mathrm{d}x > 3\int_{B_1} |u-\bar{u}|^2\, \mathrm{d}x. $$ Integrate this inequality with respect to $r$ from zero to $\frac12$; integrating the surface integrals with respect to radius gives a volume integral. You get $$ \int_{B_{1/2}}|u-\bar{u}|^2\, \mathrm{d}x > \frac32\int_{B_1} |u-\bar{u}|^2\, \mathrm{d}x. $$ But this is impossible since the integrand is positive, $B_{1/2}\subset B_1$ and $\frac32>1$. Therefore the claim is indeed true.

This observation has little to do with $u$ being a minimizer. You only need $u$ to be regular enough in order to make sense of (almost all) surface integrals and apply Fubini's theorem.