Let $D \subset \mathbf{R}^n$ be the unit ball, and $u \in C^2(D)$ be a solution of the linear elliptic PDE \begin{equation} a^{ij} D_{ij} u + b^i D_i u + cu = 0 \quad \text{in $D$}, \end{equation} where the coefficients are regular, say of class $C^d$ for some integer $d \geq 1$.

Assume that $u(0) = 0$, $Du(0) = 0$. Strong unique continuation means that $u$ has finite order of vanishing at the origin unless it vanishes identically: there exists $N > 0$ so that $r^{-n} \int_{D_r} u^2 \notin O(r^N)$ as $r \to 0$ if $u \not \equiv 0$.

Question. Does this also hold for its partial derivatives $D_k u$?

Let $D \subset \mathbf{R}^n$ be the unit ball, and $u \in C^2(D)$ be a solution of the linear elliptic PDE \begin{equation} a^{ij} D_{ij} u + b^i D_i u + cu = 0 \quad \text{in $D$}, \end{equation} where the coefficients are regular, say of class $C^d$ for some integer $d \geq 1$.

Assume that $u(0) = 0$, $Du(0) = 0$. Strong unique continuation means that $u$ has finite order of vanishing at the origin unless it vanishes identically: there exists $N > 0$ so that $r^{-n} \int_{D_r} u^2 \notin O(r^N)$ as $r \to 0$ if $u \not \equiv 0$.

Question. Under the given hypotheses, do the partial derivatives $D_k u$ of $u$ also satisfy the strong unique continuation property?