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Abstract <p> We consider one-phase free boundary problems on admissible <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(N\)</EquationSource> </InlineEquation>-dimensional piecewise smooth Riemannian complexes. We first obtain the existence of minimizers <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(u\)</EquationSource> </InlineEquation> of the Bernoulli functional. Second, we prove the local Hölder continuity of these minimizers <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(u\)</EquationSource> </InlineEquation> and the local Lipschitz continuity of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(u\)</EquationSource> </InlineEquation> away from the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\((N-2)\)</EquationSource> </InlineEquation>-skeleton. Finally, we get the nondegeneracy of the minimizers near the free boundary and show that the free boundaries are sets of locally finite perimeters. </p>

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One-Phase Free Boundary Problems on Riemannian Complexes

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • Yu Peng,
  • Hui-Chun Zhang,
  • Xi-Ping Zhu

摘要

Abstract

We consider one-phase free boundary problems on admissible \(N\) -dimensional piecewise smooth Riemannian complexes. We first obtain the existence of minimizers \(u\) of the Bernoulli functional. Second, we prove the local Hölder continuity of these minimizers \(u\) and the local Lipschitz continuity of \(u\) away from the \((N-2)\) -skeleton. Finally, we get the nondegeneracy of the minimizers near the free boundary and show that the free boundaries are sets of locally finite perimeters.