Lipschitz regularity for almost minimizers of a one-phase <i>p</i> -Bernoulli-type functional in Carnot groups of step two

Abstract In a Carnot group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔾</m:mi> </m:math> {\mathbb{G}} of step 2 and homogeneous dimension Q , we prove that almost minimizers of the (horizontal) one-phase p -Bernoulli-type functional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>J</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>u</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>:=</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:msub> <m:mrow> <m:mrow> <m:mo maxsize="120%" minsize="120%">(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mrow> <m:msub> <m:mo>∇</m:mo> <m:mi>𝔾</m:mi> </m:msub> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>p</m:mi> </m:msup> <m:mo>+</m:mo> <m:mrow> <m:msub> <m:mi>χ</m:mi> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mi>u</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo maxsize="120%" minsize="120%" rspace="4.2pt">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> see text J_{p}(u,\Omega):=\int_{\Omega}\bigl{(}|\nabla_{\mathbb{G}}u(x)|^{p}+\chi_{\{u&gt;% 0\}}(x)\bigr{)}\,dx whenever <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>&gt;</m:mo> <m:msup> <m:mi>p</m:mi> <m:mi mathvariant="normal">#</m:mi> </m:msup> <m:mo>:=</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>Q</m:mi> </m:mrow> <m:mrow> <m:mi>Q</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:mrow> </m:math> {p&gt;p^{\#}:=\frac{2Q}{Q+2}} , are locally Lipschitz continuous with respect Carnot–Carathéodory distance.