Rectifiability of solutions of the one-dimensional p-Laplacian

Abstract

In the recent papers [8] and [10] a class of Carathéodory functions ƒ(t, η, ξ) rapidly sign-changing near the boundary point t = α, has been constructed so that the equation -(|y'|p-2y')' = ƒ(t, y, y') in (α, b) admits continuous bounded solutions y whose graphs G(y) do not possess a finite length. In this paper, the same class of functions ƒ(t, η, ξ) will be given, but with slightly different input data compared to those from the previous papers, such that the graph G(y) of each solution y is a rectifiable curve in ℝ2. Moreover, there is a positive constant which does not depend on y so that length (G(y)) ≤ c < ∞.