On elliptic extensions in the disk (Q5962192)

Let \(D\) denote the unit disc, and let \(\partial u/\partial n\) denote the normal derivative of a function \(u\) at \(\partial D\). This paper is concerned with the following problem. Given arbitrary functions \(f^{(0)}\) and \(f^{(1)}\) on \(\partial D\), does there exist a function \(u\), and a linear second order, uniformly elliptic operator \(L\) with bounded measurable coefficients, such that \(Lu=0\) on \(D\) and \(u=f^{(0)}\) and \(\partial u/\partial n=f^{(1)}\) on \(\partial D\)? An affirmative answer is established under mild regularity assumptions on the boundary data. For example, it is enough to require that \(df^{(0)}/d\theta \) and \(f^{(1)}\) are Hölder continuous with exponent \(\eta >1/2\). The solution \(u\) belongs to \(L^{p}\), and has second order derivatives in \(L^{p}\), for some \(p\in (1,2)\). A similar result, where \(f^{(0)}\) and \(f^{(1)}\) are required to be analytic, had previously been established by \textit{R. Cavazzoni} [Rend. Circ. Mat. Palermo (2) 52, No. 1, 131--140 (2003; Zbl 1196.35104)].