{
  "statement": "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\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
  "starting_statement": null,
  "target_statement": "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\n\\[\n\\Delta u+k^2u=0\\quad\\text{in }\\Omega,\n\\qquad\nu=0\\quad\\text{on }\\partial\\Omega,\n\\qquad\n\\partial_\\nu u=1\\quad\\text{on }\\partial\\Omega,\n\\]\nwhere \\(\\nu\\) is the outward unit normal.",
  "case_id": "EP-1-H0-to-H5",
  "given_context": {
    "premise": null
  },
  "protocol": "single"
}