Let $u,u_k \in C^{0}(K)$and $u_0 \in C(\partial K)$ where $K \subset \mathbb{R}^{n}$ is a compact set. Assume that $u_k \rightarrow u$ uniformly and exist $x_0, x_1 \in \partial K$such that $u(x_0) < 0$ and $u(x_1)>0$.
These hypotheses are sufficient to guarantee that \begin{equation} \mbox{med}(\{u_k>0\}) \rightarrow \mbox{med}(\{u>0\}) \end{equation} or \begin{equation} \mbox{med}(\{u_k>0\}) \rightarrow \mbox{med}(\{u>0\})? \end{equation}
Is there some type of converge such $\{u_k>0\} \rightarrow \{u>0\}$ or $\{u_k>0\} \rightarrow \{u>0\}?$