Let $\Omega$ be an open set in $\mathbb{R}^n$ and $K\subset\Omega$ be a compact subset. Assume $\{f_i\}$ is a sequence of solutions of the equation $\Delta f_i=0$ on $\Omega$, with $\|f_i\|_{L^2(\Omega)}\leq C$ for some constant number $C$ independent of $i$.
Question: does the inequality $$ \|f_i\|_{L^{\infty}(K)}\leq \lambda $$ hold for some constant number $\lambda$? (Of coursewe have that $\|f_i\|_{L^{\infty}(K)}=\max_{K}|f_i(x)|$ due to the continuity of $f_i$)
Since $K$ is compact, there is $\delta >0$ so that for all $y\in K$, the ball $B_y(\delta)$ lies in $U$.
Let $u$ be a harmonic functions on $\Omega$. For each $x\in K$, by the mean value property for harmonic functions,
$$ u(x) = \frac{1}{|B_x(\delta)|} \int_{B_x(\delta)} u(y) dy$$
Thus by Cauchy-Schwarz,
\begin{align} |u(x)| &\le \frac{1}{|B_x(\delta)|} \int_{B_x(\delta)} |u(y)| dy \\ &\le \frac{1}{|B_x(\delta)|} \|u\|_{L^2(B_x(\delta))} \| 1\|_{L^2(B_x(\delta))} \\ &\le \frac{1}{\sqrt{|B_x(\delta)|}} \|u\|_{L^2(\Omega)}. \end{align}
Thus $$\sup |u| \le \frac{1}{\sqrt{\omega_n}\delta^{n/2}} \| u\|_{L^2(\Omega)}, $$
here $\omega_n$ is the volumn of $B_0(1)$.