On the 2-dimensional dual Minkowski problem

We study the 2-dimensional dual Minkowski problem, which is the following nonlinear problem on unit circleu″+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⋯}.