# Berenstein rigidity — H4

Let \(D=\{z\in\mathbb C:|z|<1\}\). There exist an integer \(m\ge2\), a real number \(k>0\), a holomorphic function \(\phi\) on a neighborhood of \(\overline D\), and a real-valued function \(U\in C^2(\overline D)\), such that
\[
\phi(0)=0,\qquad \phi'(0)=1,\qquad
\operatorname{Re}\phi'(z)>0\quad(z\in\overline D),
\]
\[
\phi(e^{2\pi i/m}z)=e^{2\pi i/m}\phi(z)\quad(z\in D),
\qquad \phi\not\equiv z,
\]
and
\[
\Delta U+k^2|\phi'|^2U=0\quad\text{in }D,
\]
\[
U=0,\qquad \partial_r U=|\phi'|\quad\text{on }\partial D,
\]
where \(\partial_r\) is the outward radial derivative on the unit circle.