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<p>Given a real-valued function having a nondegenerate compact manifold of critical points, some of these points survive under small <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(C^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-perturbations. This is a well-known result in critical point theory. In 1986, Weinstein obtained the analogous conclusions when the perturbation is only <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(C^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> and the ambient space is a finite-dimensional manifold. In this work we present a complete proof for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(C^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> perturbations in infinite-dimensional Hilbert spaces.</p>

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\(C^1\) perturbations of a continuum of critical points

  • 期刊论文

  • 发表日期：

    2025年12月13日

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  • R. Ortega,
  • A. J. Ureña

摘要

Given a real-valued function having a nondegenerate compact manifold of critical points, some of these points survive under small \(C^2\) C 2 -perturbations. This is a well-known result in critical point theory. In 1986, Weinstein obtained the analogous conclusions when the perturbation is only \(C^1\) C 1 and the ambient space is a finite-dimensional manifold. In this work we present a complete proof for \(C^1\) C 1 perturbations in infinite-dimensional Hilbert spaces.