Let $B \subset \mathbb{R}^d$ be the ball of radius one, and consider the map defined on $L^2(B,\mathbb{R})$ \begin{align*} f(\phi) = \underset{\substack{ \varphi \in H^1(B,\mathbb{R}) \\ \left< \varphi,\phi \right> = 0 \\ \int \left|\varphi\right|^2=1}}{\text{min}} \int \left|\nabla \varphi\right|^2 \end{align*} Let $\phi_n \in L^2(B,\mathbb{R})$ be a sequence with $\int \left|\phi_n\right|^2=1$ converging to $\phi$ weakly in $L^2(B)$, do we have $\limsup_{n \rightarrow +\infty} f(\phi_n) \le f(\phi)$ ?


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