Electron. J. Diff. Eqns., Vol. 2005(2005), No. 46, pp. 1-8.

Rectifiability of solutions of the one-dimensional p-Laplacian

Mervan Pasic

In the recent papers [8] and [10] a class of Caratheodory 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 $-(|y'|^{p-2}y')'=f(t,y,y')$ 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 $\mathop{\rm length}(G(y))\leq c$ less than $\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.

Show me the PDF file (235K), TEX file, and other files for this article.

Mervan Pasic
Department of Mathematics
Faculty of Electrical Engineering and Computing
University of Zagreb
Unska 3, 10000 Zagreb, Croatia
email: mervan.pasic@fer.hr

Return to the EJDE web page