Lower bound of surface integral of harmonic functions?

Question

I think the following is true, but I cannot find a proof or reference for it.
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with smooth, connected boundary. Let $a_{ij},\ 1\leq i,j\leq n$ be continuous functions on $\partial\Omega$ such that $a_{ij}=a_{ji}.$ If there is a $\delta>0$ such that (where $dS$ is the surface measure, $d\lambda$ is the Lebesgue measure) $$\sum_{i,j=1}^n\int_{\partial\Omega}a_{ij}f_if_jdS\geq \delta\sum_{i=1}^n\int_{\Omega}f_i^2d\lambda$$ for all harmonic functions $f_i,$ then the matrix $(a_{ij})$ is positive definite.