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<p>Given an <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we prove that stationary two-valued harmonic functions with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Hölder regularity are locally Lipschitz continuous. Our method relies on Almgren’s monotonicity formula and the analysis of blow-up limit. This result is applicable to Almgren’s 2-valued functions which are stationary with respect to outer and inner variations.</p>

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Lipschitz Regularity for Stationary Two-Valued Harmonic Functions Under Hölder Assumption

  • 期刊论文

  • 发表日期：

    2026年5月11日

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  • Lingxiao Cheng

摘要

Given an \(\alpha \in (0,1)\) α ∈ ( 0 , 1 ) , we prove that stationary two-valued harmonic functions with \(\alpha \) α -Hölder regularity are locally Lipschitz continuous. Our method relies on Almgren’s monotonicity formula and the analysis of blow-up limit. This result is applicable to Almgren’s 2-valued functions which are stationary with respect to outer and inner variations.