On the unique continuation property of elliptic divergence form equations in the plane (Q1127786)

We are concerned with the local behavior of weak solutions \(\varphi\in W^{1,2} (\Omega)\) of linear elliptic equations of divergence structure in a plane domain \(\Omega \subset \mathbb{R}^2\). We study equations of the form \[ L\varphi =-D_\alpha \bigl(a^{\alpha \beta} (z) D_\beta \varphi +b^\alpha (z)\varphi \bigr) +c^\alpha (z)D_\alpha \varphi+ d(z)\varphi=0, \tag{1} \] whose coefficients \(a^{\alpha \beta}, b^\alpha, c^\alpha, d\) \((\alpha, \beta=1,2)\) are only assumed to be measurable and essentially bounded functions on \(\Omega\). We assume that the operator \(L\) is uniformly elliptic in \(\Omega\). We study the local behavior of \(\varphi\) near the origin. We prove the following unique continuation property: Suppose that \(\varphi= o(| z|^n)\) as \(| z|\to 0\) for all \(n\in\mathbb{N}\). Then \(\varphi \equiv 0\).