# Berenstein rigidity EP-1-H0-to-H5

## Setting

You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.

## Problem

Prove H5: There exist a bounded simply connected domain \(\Omega\subset\mathbb R^2\) with \(C^\infty\) boundary, a real number \(k>0\), and a nonzero real-valued function \(u\in C^2(\overline\Omega)\), such that \(\Omega\) is not a disk and
\[
\Delta u+k^2u=0\quad\text{in }\Omega,
\qquad
u=0\quad\text{on }\partial\Omega,
\qquad
\partial_\nu u=1\quad\text{on }\partial\Omega,
\]
where \(\nu\) is the outward unit normal.

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.