Solution to the Khavinson problem near the boundary of the unit ball (Q524410)

The paper deals with pointwise sharp estimates for the gradient of real-valued bounded harmonic functions. More precisely, let \(n>2\) and let \(\mathbb{B}_n\) denote the unit ball in \(\mathbb{R}^n\). For every point \(x\) in \(\mathbb{B}_n\) and for every \(l \in \partial \mathbb{B}_n\), let \(\mathcal{C}(x)\) denote the optimal number for the gradient estimate \[ |\nabla U(x)| \leq \mathcal{C}(x)\sup_{y} |U(y)| \] and let \(\mathcal{C}(x;l)\) denote the optimal number for the gradient estimate in direction \(l\), i.e., the smallest number such that \[ |\langle\nabla U(x), l\rangle| \leq \mathcal{C}(x; l)\sup_{y} |U(y)|\, . \] Here \(U\) is a bounded harmonic function in \(\mathbb{B}_n\). The author shows that \[ \mathcal{C}(x)=\mathcal{C}(x; \mathrm n_x)\, , \] when \(x\) is close to \(\partial \mathbb{B}_n\), where \(\mathrm n_x\) denotes the normal direction at the point \(x\). This result partially confirms a conjecture by \textit{D. Khavinson} [Can. Math. Bull. 35, No. 2, 218--220 (1992; Zbl 0776.31004)].