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<p>In this paper we prove a classification result for axially symmetric one-phase stable solutions of the Alt-Phillips free boundary problem in the range <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\gamma \in (0, 2/3)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">/</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We show that such solutions are one-dimensional in dimensions 3, 4, and 5 for a range of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\gamma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> depending on the dimension. To accomplish this, we establish a stability inequality that extends the one for the Alt-Caffarelli problem (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\gamma = 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>) to the whole range <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\gamma \in (0,2)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Stable cones in the Alt-Phillips free boundary problem

  • 期刊论文

  • 发表日期：

    2026年2月17日

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  • Aram Karakhanyan,
  • Tomás Sanz-Perela

摘要

In this paper we prove a classification result for axially symmetric one-phase stable solutions of the Alt-Phillips free boundary problem in the range \(\gamma \in (0, 2/3)\) γ ∈ ( 0 , 2 / 3 ) . We show that such solutions are one-dimensional in dimensions 3, 4, and 5 for a range of \(\gamma \) γ depending on the dimension. To accomplish this, we establish a stability inequality that extends the one for the Alt-Caffarelli problem ( \(\gamma = 0\) γ = 0 ) to the whole range \(\gamma \in (0,2)\) γ ∈ ( 0 , 2 ) .