On solvability of two-dimensional Lp-Minkowski problem

Given p not equal 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) = integral(0)(T) { (u' (x))(2) - u(2)(x) }dx + C(integral(0)(T) g(x)u(p)(x) dx)(2/p) has a minimizer in the class of positive functions u is an element of C-1 (R) for which u(x + T) = u(x) for all x is an element of R. This minimizer is used to prove the existence of a positive periodic solution y is an element of C-2 (R) of two-dimensional L-p-Minkowski problem y(1-p)(x)(y"(x) + y(x)) = g(x), where p is not an element of {0, 2}. (C) 2003 Elsevier Science (USA). All rights reserved.