  Let $\Omega \subset \mathbb{R}^d$ be a quasiconvex Lipschitz domain and
$A(x)$ be a $d \times d$ uniformly elliptic, symmetric matrix with Lipschitz
coefficients. Assume a nontrivial $u$ solves $-\nabla \cdot (A(x) \nabla u) =
0$ in $\Omega$, and $u$ vanishes on $\Sigma = \partial \Omega \cap B$ for some
ball $B$. The main contribution of this paper is to demonstrate the existence
of a countable collection of open balls $(B_i)_i$ such that the restriction of
$u$ to $B_i \cap \Omega$ maintains a consistent sign. Furthermore, for any
compact subset $K$ of $\Sigma$, the set difference $K \setminus \bigcup_i B_i$
is shown to possess a Minkowski dimension that is strictly less than $d - 1 -
\epsilon$. As a consequence, we prove Lin's conjecture in quasiconvex domains.
