EDIT: Basically, I need to find some relationship between the partial derivatives of the functions $\theta$ and the functionals $Q_n$.
Let $\mathcal{H} \subset \mathbb{R}^3$ denote a compact subspace. Suppose we have a sequence of functionals $(Q_n)_{n\geq 1}$ and a functional $Q$ from $C(\mathcal{H},\mathbb{R}^3)$ (which is the space of continuous functions from $\mathcal{H}$ to $\mathbb{R}^3)$ to $\mathbb{R}$, thus $$Q:C(\mathcal{H}) \rightarrow \mathbb{R}$$ and $$Q_n:C(\mathcal{H}) \rightarrow \mathbb{R}.$$
Suppose $Q_n\rightarrow Q$ uniformly in $C(\mathcal{H},\mathbb{R}^3)$ and that $Q$ has a unique minimum $\theta_0 \in C(\mathcal{H},\mathbb{R}^3)$.
I have two questions which arise from wanting to prove that the minimizer of $Q_n$ converges to $\theta_0$ (we use the sup norm $\|\cdot\|_\infty$).
Does the minimizer of $Q_n$ converges to $\theta_0$?
What conditions do I need on $Q_n$ if I want the sequence of minimizers of $Q_n$ to have uniformly bounded first order partial derivatives?