I have a complex valued function on the boundary of a set, and want to know if it has a holomorphic extension to the entire set (holomorphic in interior, continuous up to the boundary).
In other words: Given a smooth open subset $\Omega$ of $\mathbb{C}$, and a smooth complex-valued function $f:\partial\Omega \to \mathbb{C}$, when does there exist a function $u \in C(\bar{\Omega})$ which is holomorphic in the interior and such that $u(z)=f(z)$ for any $z \in \partial\Omega$?
This came up in research. I'm studying a free boundary problem on the region $\{(x,y)\in \mathbb{R}^2|y\leq h(x)\}$ (assume $h$ smooth to keep things simple) and have a vector field $(x,h(x)) \mapsto [1,h'(x)]$ which is tangent to the free surface. It would be very helpful if I could extend that vector field to the entire domain (or at least some subdomain, e.g. $\{(x,y)\in \mathbb{R}^2|y\in (h(x)-\varepsilon,h(x)]\}$) while satisfying the Cauchy-Riemann equations. Unfortunately I haven't thought about complex analysis in about a decade.