Rectifiability of solutions of the one-dimensional p-Laplacian
Mervan Pašić · DOAJ (DOAJ: Directory of Open Access Journals) · 2005
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 $mathop{m length}(G(y))leq c$ less than $infty $.