Let $B_R$ a ball in $\mathbb{R}^n$. Consider $u^{\star} \in H^{1}(B_R) $ and $f \in H^{1}(B_R) \cap C(\overline{B_R})$. Suppose that $u^{\star}$ minimizes $$\int_{B_R} |\nabla u|^2, u \in \{ v \in H^{1}(B_R); v - f \in H^{1}_{0}(B_R)\} .$$
Let $u^{\sharp} \in C^{2}(B_R) \cap C(\overline{B_R})$ such that $\Delta u^{\sharp} = 0 , $ in $B_R$ (pointwise) and $u=f$ in $\partial B_R$ (pointwise). It is reasonable to expect that $u^{\sharp} = u^{\star}$. I am not seeing how to prove this, someone could give me help?