{
  "statement": "H2: Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\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\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\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.\n\nProve 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": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\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\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\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.",
  "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-H2-to-H5",
  "given_context": {
    "premise": "Let \\(D=\\{z\\in\\mathbb C:|z|<1\\}\\). Let \\(\\phi\\) be holomorphic on a neighborhood of \\(\\overline D\\), with\n\\[\n\\phi(0)=0,\\qquad \\phi'(0)=1,\\qquad\n\\operatorname{Re}\\phi'(z)>0\\quad(z\\in\\overline D).\n\\]\nSuppose \\(k>0\\) and a real-valued function \\(U\\in C^2(\\overline D)\\) satisfy\n\\[\n\\Delta U+k^2|\\phi'|^2U=0\\quad\\text{in }D,\n\\]\n\\[\nU=0,\\qquad \\partial_r U=|\\phi'|\\quad\\text{on }\\partial D,\n\\]\nwhere \\(\\partial_r\\) is the outward radial derivative on the unit circle.\n\nThen \\(\\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\n\\[\nu=U\\circ\\phi^{-1}\n\\]\nis a nonzero real-valued function in \\(C^2(\\overline\\Omega)\\) satisfying\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."
  },
  "protocol": "single"
}