<p>We study the regularity of the interface between the disjoint supports of a pair of nonnegative subharmonic functions. The portion of the interface where the Alt–Caffarelli–Friedman (ACF) monotonicity formula is asymptotically positive forms an <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2024_3077_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}^{n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-rectifiable set. Moreover, for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2024_3077_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}^{n-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">H</mi> </mrow> <mrow> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>-a.e. such point, the two functions have unique blowups, i.e. their Lipschitz rescalings converge in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2024_3077_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="35" /> </InlineMediaObject> <EquationSource Format="TEX">$W^{1,2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>W</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> to a pair of nondegenerate truncated linear functions whose supports meet at the approximate tangent plane. The main tools used include the Naber–Valtorta framework and our recent result establishing a sharp quantitative remainder term in the ACF monotonicity formula. We also give applications of our results to free boundary problems.</p>

Rectifiability and uniqueness of blow-ups for points with positive Alt–Caffarelli–Friedman limit

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2025年1月6日

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Mark Allen,
Dennis Kriventsov,
Robin Neumayer

摘要

We study the regularity of the interface between the disjoint supports of a pair of nonnegative subharmonic functions. The portion of the interface where the Alt–Caffarelli–Friedman (ACF) monotonicity formula is asymptotically positive forms an $\mathcal {H}^{n-1}$ $H^{n - 1}$ -rectifiable set. Moreover, for $\mathcal {H}^{n-1}$ $H^{n - 1}$ -a.e. such point, the two functions have unique blowups, i.e. their Lipschitz rescalings converge in $W^{1,2}$ $W^{1, 2}$ to a pair of nondegenerate truncated linear functions whose supports meet at the approximate tangent plane. The main tools used include the Naber–Valtorta framework and our recent result establishing a sharp quantitative remainder term in the ACF monotonicity formula. We also give applications of our results to free boundary problems.