On the 2-dimensional dual Minkowski problem

Abstract

We study the 2-dimensional dual Minkowski problem, which is the following nonlinear problem on unit circle u” + u = g(θ)u−1(u2+u’2)(2−k)/2, θ ∈ S for any given positive continuous function g(θ) with 2π/m-periodic. We prove that it is solvable for all k ∈ (1, +∞) and m ∈ {3,4,5···}.