Let $f:\mathbb{S}^1 \to \mathbb{C}$ mathbb{S}^1$be a continuous functionan oreintation-preserving homeomorphism. Denote by$H(f)$the complex harmonic extension/solution in$\mathbb{D}$to the Dirichlet problem with boundary data$f$. It f$, 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 $!$

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# What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases ?

Let $f:\mathbb{S}^1 \to \mathbb{C}$ be a continuous function. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data $f$.

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 $!$