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<p>This paper presents a conditional convergence result of solutions to the Allen–Cahn equation with multiwell potentials to a De Giorgi-type <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( {{\,\textrm{BV}\,}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.166667em" /> <mtext>BV</mtext> <mspace width="0.166667em" /> </mrow> </math></EquationSource> </InlineEquation>-solution to multiphase mean curvature flow. Moreover, we show that De Giorgi-type <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( {{\,\textrm{BV}\,}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mspace width="0.166667em" /> <mtext>BV</mtext> <mspace width="0.166667em" /> </mrow> </math></EquationSource> </InlineEquation>-solutions are De Giorgi-type varifold solutions, and thus our solution is unique in a weak-strong sense.</p>

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Convergence of Allen–Cahn equations to De Giorgi’s multiphase mean curvature flow

  • 期刊论文

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  • 发表日期：

    2026年5月7日

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  • Pascal Steinke

摘要

This paper presents a conditional convergence result of solutions to the Allen–Cahn equation with multiwell potentials to a De Giorgi-type \( {{\,\textrm{BV}\,}}\) BV -solution to multiphase mean curvature flow. Moreover, we show that De Giorgi-type \( {{\,\textrm{BV}\,}}\) BV -solutions are De Giorgi-type varifold solutions, and thus our solution is unique in a weak-strong sense.