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

In a Carnot group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔾</m:mi> </m:math> {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>></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 Jₚ(u,Ω):=∫Ω{(}|∇Gu(x)|ᵖ+χ_{\{u>% 0\}}(x){)}\,dx whenever <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></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>p#:=2Q/Q+2} , are locally Lipschitz continuous with respect Carnot–Carathéodory distance.