A uniqueness result for harmonic functions (Q2781341)

Let \(D\subset\mathbb{R}^{d+1}\) be the half space \(x_{d+1}>0\), \(d\geq 2\). Suppose \(u\) is harmonic on \(D\) and continuous on \(\overline{D}\). The author defines a certain matrix \(\widetilde a(x)\), \(x\in D\), in terms of the second order partial derivatives of \(u\). The matrix \(\widetilde a\) is related to a certain martingale defined in terms of the first order partial derivatives of \(u\). The author proves that if there exists a subset \(A\) of \(\partial D\) of positive \(d\)-dimensional Lebesgue measure where the gradient of \(u\) vanishes continuously and such that \(\widetilde a\) has a continuous extension of rank at least three at all points of \(A\), then \(u\) is constant.