Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 46, pp. 1-8.
Title: Rectifiability of solutions of the
one-dimensional p-Laplacian
Author: Mervan Pasic (Univ. of Zagreb, Croatia)
Abstract:
In the recent papers [8] and [10] a class of
Carath\'{e}odory functions $f(t,\eta ,\xi )$ rapidly sign-changing
near the boundary point $t=a$, has been constructed so that the
equation $-(|y'|^{p-2}y')'=f(t,y,y')$ in $(a,b)$ admits continuous
bounded solutions $y$ whose graphs $G(y)$ do not possess a finite
length. In this paper, the same class of functions $f(t,\eta ,\xi )$
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 $\mathbb{R}^{2}$. Moreover,
there is a positive constant which does not depend on $y$ so that
$\mathop{\rm length}(G(y))\leq c<\infty $.
Submitted November 11, 2004. Published April 24, 2005.
Math Subject Classifications: 35J60, 34B15, 28A75.
Key Words: Nonlinear p-Laplacian; bounded continuous solutions;
graph; qualitative properties; length; rectifiability.