Let $u\in H^1(B_1;S^k)$, where $$B_1:= \{x\in\mathbb{R}^n: \lvert x\rvert<1\}\\ S^k:=\{x\in \mathbb{R}^{k+1}: \lvert x\rvert=1\}. $$ Suppose $u$ is a minimizer for the Dirichlet energy functional $$ I[u]:=\int_{B_1}\lvert Dw\rvert^2\mathrm{d}x $$ We denote $\bar{u}:=\frac{1}{|{B_1}|}\int_{B_1}u\mathrm{d}x$. Let $\rho\in (\frac{1}{2},1)$ such that
$$ \int\limits_{\partial B_\rho}|u-\bar{u}|^2\mathrm{d}x\le 3\int\limits_{B_1}|u-\bar{u}|^2\mathrm{d}x $$
$$ \int\limits_{\partial B_{\rho}}|Du|^2\mathrm{d}x\le 3\int\limits_{B_1}|Du|^2\mathrm{d}x. $$
Let $v:B_{\rho}\to\mathbb{R}^{k+1}$ such that $$ \begin{cases} \Delta v=0\quad\text{in}\;\, B_\rho \\ v=u\quad\text{on}\;\,\partial B_\rho \end{cases}. $$ I have to prove that $$ \int\limits_{B_\rho}|Dv|^2\mathrm{d}x\le c(n)(\int\limits_{\partial B_{\rho}}|D_Tu|^2\mathrm{d}x\int\limits_{\partial B_{\rho}}|u-\bar{u}|^2\mathrm{d}x)^{1/2} $$ where $D_Tu$ is the tangential derivative of $u$.