On solvability of two-dimensional Lp-Minkowski problem

Abstract

Given p≠0 and a positive continuous function g, with g( x+ T)= g( x), for some 0< T<1 and all real x, it is shown that for suitable choice of a constant C>0 the functional F(u)= ∫ 0 T {(u′(x)) 2−u 2(x)}dx+C( ∫ 0 T g(x)u p(x) dx) 2/p has a minimizer in the class of positive functions u∈ C 1( R) for which u( x+ T)= u( x) for all x∈ R. This minimizer is used to prove the existence of a positive periodic solution y∈ C 2( R) of two-dimensional L p -Minkowski problem y 1− p ( x)( y″( x)+ y( x))= g( x), where p∉{0,2}.