Question

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an oreintation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data $f$, which can also be shown to be a homeomorphism of $\bar{\mathbb{D}}$ [Choquet's theorem].

Now,it follows from a well-known theorem of Kellog-Warschowski that if $f\in C^{k,\alpha}(\mathbb{S^1})$, then $H(f) \in C^{k,\alpha}(\mathbb{D})$.

My question is : assuming $k=1$, what is the limiting matrix of the (total) derivative matrix $DH(f)_p$ as $p\to \zeta \in \mathbb{S^1},p \in \mathbb{D}$ ? From some other kind of extensions I have seen before, I would take a guess that $DH(f)_p \to f'(\zeta).Id$ as $p\to \zeta$, where $Id$ denotes the $2$ by $2$ identity matrix.

A reference to your answer will be highly appreciated $!$

Answer 1

Your guess is wrong. Take real $f$, for example, then harmonic extension is also real, and its derivative is singular. In general, I do not expect a simple answer. And certainly the answer cannot depend on the local properties of $f$ near $\zeta$ only. Indeed, leave $f$ unchanged on one half of the circle, but change it on another half. It is clear that the derivative at the points of the first half will change. So the limit of the derivative at a point cannot be computed from the knowledge of f only near this point.