Classification of solutions for the planar isotropic 𝐿 𝑝 dual Minkowski problem

In [B. Andrews, Classification of limiting shapes for isotropic curve flows, J. Amer. Math. Soc. 16 (2003), 2, 443–459], Ben Andrews obtained the complete classification of the solutions of the planar isotropic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> Lₚ Minkowski problem. In this paper, by generalizing his result, we obtain the complete classification of the solutions of the planar isotropic <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> Lₚ dual Minkowski problem for any <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> </m:math> p,q . That is, we classify all embedded solutions of the equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msup> <m:mi>u</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msubsup> <m:mi>u</m:mi> <m:mi>θ</m:mi> <m:mn>2</m:mn> </m:msubsup> <m:mo>+</m:mo> <m:msup> <m:mi>u</m:mi> <m:mn>2</m:mn> </m:msup> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mfrac> <m:mrow> <m:mi>q</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mn>2</m:mn> </m:mfrac> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mrow> <m:mi>θ</m:mi> <m:mo>⁢</m:mo> <m:mi>θ</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mspace width="1em"/> <m:mrow> <m:mtext>on</m:mtext> <m:mo lspace="0.500em">⁢</m:mo> <m:msup> <m:mi mathvariant="double-struck">S</m:mi> <m:mn>1</m:mn> </m:msup> </m:mrow> </m:mrow> </m:mrow> <m:mo lspace="0em">.</m:mo> </m:mrow> </m:math> u¹⁻ᵖ(uθ²+u²)q-2/2(uθθ+u)=1\ S¹. To establish the classification, we convert the ODE for the solution into an integral and study its asymptotic behavior, duality and monotonicity.