Jacobian determinants for nonlinear gradient of planar ∞-harmonic functions and applications

We introduce a distributional Jacobian determinant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo rspace="0.167em">det</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>V</m:mi> <m:mi>β</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> DVβ(Dv) in dimension two for the nonlinear complex gradient <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>V</m:mi> <m:mi>β</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>β</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>v</m:mi> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> </m:msub> <m:mo>,</m:mo> <m:mrow> <m:mo>−</m:mo> <m:msub> <m:mi>v</m:mi> <m:msub> <m:mi>x</m:mi> <m:mn>2</m:mn> </m:msub> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> Vβ(Dv)= Dvβ(v_{x₁},-v_{x₂}) for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>β</m:mi> <m:mo>></m:mo> <m:mrow> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> β>-1 , whenever <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>v</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mi>W</m:mi> <m:mi>loc</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:math> v∈ W^{1{,}2}loc and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>β</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>v</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>β</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>∈</m:mo> <m:msubsup> <m:mi>W</m:mi> <m:mi>loc</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msubsup> </m:mrow> </m:math> β Dv1+β∈ W^{1{,}2}loc . This is new when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>β</m:mi> <m:mo>≠</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> β≠ 0 . Given any planar ∞-harmonic function 𝑢, we show that such distributional Jacobian determinant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo rspace="0.167em">det</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>V</m:mi> <m:mi>β</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> DVβ(Du) is a nonnegative Radon measure with some quantitative local lower and upper bounds. We also give the following two applications. Applying this result with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>β</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> β=0 , we develop an approach to build up a Liouville theorem, which improves that of Savin. Precisely, if 𝑢 is an ∞-harmonic function in the whole <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mn>2</m:mn> </m:msup> </m:math> R² with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:munder> <m:mo movablelimits="false">lim inf</m:mo> <m:mrow> <m:mi>R</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:munder> <m:mrow> <m:munder> <m:mo lspace="0.167em" movablelimits="false" rspace="0.167em">inf</m:mo> <m:mrow> <m:mi>c</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> </m:munder> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mi>R</m:mi> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:msub> <m:mo rspace="0em">⨍</m:mo> <m:mrow> <m:mi>B</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>R</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mrow> <m:mi>u</m:mi> <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>−</m:mo> <m:mi>c</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mo lspace="0.170em">⁢</m:mo> <m:mrow> <m:mo mathvariant="italic" rspace="0em">d</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo><</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> R→∞infc1/RB(0,R) u(x)-c\,dx<∞, then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mi>b</m:mi> <m:mo>+</m:mo> <m:mrow> <m:mi>a</m:mi> <m:mo lspace="0.222em" rspace="0.222em">⋅</m:mo> <m:mi>x</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:math> u=b+a· x for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>b</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> </m:math> b and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>a</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi mathvariant="double-struck">R</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> </m:math> a² . Denoting by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>u</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> uₚ the 𝑝-harmonic function having the same nonconstant boundary condition as 𝑢, we show that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mo rspace="0.167em">det</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>V</m:mi> <m:mi>β</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>u</m:mi> <m:mi>p</m:mi> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo rspace="0.1389em" stretchy="false">→</m:mo> <m:mrow> <m:mo lspace="0.1389em" rspace="0.167em">det</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>V</m:mi> <m:mi>β</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mrow> </m:math> DVβ(Duₚ)→ DVβ(Du) as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo stretchy="false">→</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> p→∞ in the weak-⋆ sense in the space of Radon measure. Recall that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>V</m:mi> <m:mi>β</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:msub> <m:mi>u</m:mi> <m:mi>p</m:mi> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Vβ(Duₚ) is always quasiregular mappings, but <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>V</m:mi> <m:mi>β</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>D</m:mi> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Vβ(Du) is not in general.