# Berenstein rigidity — H2

Let \(D=\{z\in\mathbb C:|z|<1\}\). Let \(\phi\) be holomorphic on a neighborhood of \(\overline D\), with
\[
\phi(0)=0,\qquad \phi'(0)=1,\qquad
\operatorname{Re}\phi'(z)>0\quad(z\in\overline D).
\]
Suppose \(k>0\) and a real-valued function \(U\in C^2(\overline D)\) satisfy
\[
\Delta U+k^2|\phi'|^2U=0\quad\text{in }D,
\]
\[
U=0,\qquad \partial_r U=|\phi'|\quad\text{on }\partial D,
\]
where \(\partial_r\) is the outward radial derivative on the unit circle.

Then \(\phi\) is injective on a neighborhood of \(\overline D\). The set \(\Omega=\phi(D)\) is a bounded simply connected domain with real-analytic Jordan boundary, and
\[
u=U\circ\phi^{-1}
\]
is a nonzero real-valued function in \(C^2(\overline\Omega)\) satisfying
\[
\Delta u+k^2u=0\quad\text{in }\Omega,
\qquad
u=0\quad\text{on }\partial\Omega,
\qquad
\partial_\nu u=1\quad\text{on }\partial\Omega,
\]
where \(\nu\) is the outward unit normal.