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<p>We discuss <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\((K,\!N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="-0.166667em" /> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-convexity and gradient flows for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\((K,\!N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="-0.166667em" /> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-convex functionals on metric spaces, in the case of real&#xa0;<i>K</i> and <i>negative</i>&#xa0;<i>N</i>. In this generality, it is necessary to consider functionals unbounded from below and/or above, possibly attaining as values both the positive and the negative infinity. We prove several properties of gradient flows of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\((K,\!N)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mo>,</mo> <mspace width="-0.166667em" /> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-convex functionals characterized by Evolution Variational Inequalities, including contractivity, regularity, and uniqueness.</p>

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Gradient flows of \((K,\!N)\)-convex functions with negative N

  • 期刊论文

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  • 发表日期：

    2026年2月17日

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  • Lorenzo Dello Schiavo,
  • Mattia Magnabosco,
  • Chiara Rigoni

摘要

We discuss \((K,\!N)\) ( K , N ) -convexity and gradient flows for \((K,\!N)\) ( K , N ) -convex functionals on metric spaces, in the case of real K and negative N. In this generality, it is necessary to consider functionals unbounded from below and/or above, possibly attaining as values both the positive and the negative infinity. We prove several properties of gradient flows of \((K,\!N)\) ( K , N ) -convex functionals characterized by Evolution Variational Inequalities, including contractivity, regularity, and uniqueness.