<p>We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\textrm{Per}_{w}(E)=\int _{\partial ^*E}w\,\textrm{d}\mathcal {H}^{n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>Per</mtext> <mi>w</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow> <msup> <mi>∂</mi> <mo>∗</mo> </msup> <mi>E</mi> </mrow> </msub> <mi>w</mi> <mspace width="0.166667em" /> <mtext>d</mtext> <msup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, where <i>w</i> is a weight asymptotic to the distance function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$d(\cdot ,\mathbb {R}\setminus \Omega )^a$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo>,</mo> <mi mathvariant="double-struck">R</mi> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mi>a</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> near <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. This implies that the boundaries of almost-minimizers are <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$C^{1,\gamma _0}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <msub> <mi>γ</mi> <mn>0</mn> </msub> </mrow> </msup> </math></EquationSource> </InlineEquation>-surfaces that touch <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\partial \Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi mathvariant="normal">Ω</mi> </mrow> </math></EquationSource> </InlineEquation> orthogonally, up to a singular set <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textrm{Sing}(\partial E)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Sing</mtext> <mo stretchy="false">(</mo> <mi>∂</mi> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> whose Hausdorff dimension satisfies the bound <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\text {dim}_\mathcal {H}(\textrm{Sing}(\partial E)) \le n +a -(5+\sqrt{8})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>dim</mtext> <mi mathvariant="script">H</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mtext>Sing</mtext> <mrow> <mo stretchy="false">(</mo> <mi>∂</mi> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mi>n</mi> <mo>+</mo> <mi>a</mi> <mo>-</mo> <mrow> <mo stretchy="false">(</mo> <mn>5</mn> <mo>+</mo> <msqrt> <mn>8</mn> </msqrt> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Regularity for free boundary surfaces minimizing degenerate area functionals

期刊论文
发表日期：

2026年6月17日

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Carlo Gasparetto,
Filippo Paiano,
Bozhidar Velichkov

摘要

We establish an epsilon-regularity theorem at points in the free boundary of almost-minimizers of the energy $\textrm{Per}_{w}(E)=\int _{\partial ^*E}w\,\textrm{d}\mathcal {H}^{n-1}$ ${Per}_{w} (E) = \int_{\partial^{*} E} w d H^{n - 1}$ , where w is a weight asymptotic to the distance function $d(\cdot ,\mathbb {R}\setminus \Omega )^a$ $d {(\cdot, R \ Ω)}^{a}$ near $\partial \Omega $ $\partial Ω$ and $a>0$ $a > 0$ . This implies that the boundaries of almost-minimizers are $C^{1,\gamma _0}$ $C^{1, γ_{0}}$ -surfaces that touch $\partial \Omega $ $\partial Ω$ orthogonally, up to a singular set $\textrm{Sing}(\partial E)$ $Sing (\partial E)$ whose Hausdorff dimension satisfies the bound $\text {dim}_\mathcal {H}(\textrm{Sing}(\partial E)) \le n +a -(5+\sqrt{8})$ ${dim}_{H} (Sing (\partial E)) \leq n + a - (5 + \sqrt{8})$ .