<p>We prove that nonnegative almost minimizers of the horizontal Bernoulli-type functional <Equation ID="Equ100"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="526_2025_2959_Article_Equ100.gif" Format="GIF" Height="41" Rendition="HTML" Resolution="72" Type="Linedraw" Width="301" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} J(u,\Omega ):=\int _{\Omega }\Big (|\nabla _{\mathbb {G}}u(x)|^2+\chi _{\{u>0\}}(x)\Big )\,dx \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>J</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <msub> <mo>∫</mo> <mi mathvariant="normal">Ω</mi> </msub> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <msup> <mrow> <mo stretchy="false">|</mo> <msub> <mi mathvariant="normal">∇</mi> <mi mathvariant="double-struck">G</mi> </msub> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>χ</mi> <mrow> <mo stretchy="false">{</mo> <mi>u</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mspace width="0.166667em" /> <mi>d</mi> <mi>x</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>are Lipschitz continuous with respect to the Carnot-Carathéodory distance.</p>

Regularity for almost minimizers of a one-phase Bernoulli-type functional in Carnot groups of step two

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2025年3月24日

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Fausto Ferrari,
Nicolò Forcillo,
Enzo Maria Merlino

摘要

We prove that nonnegative almost minimizers of the horizontal Bernoulli-type functional $\begin{aligned} J(u,\Omega ):=\int _{\Omega }\Big (|\nabla _{\mathbb {G}}u(x)|^2+\chi _{\{u>0\}}(x)\Big )\,dx \end{aligned}$ $\begin{matrix} J (u, Ω) : = \int_{Ω} ({| \nabla_{G} u (x) |}^{2} + χ_{{u > 0}} (x)) d x \end{matrix}$ are Lipschitz continuous with respect to the Carnot-Carathéodory distance.