On the minimum of certain functional related to the Schr?dinger equation

摘要

We consider the infimum $inf_f max limits_{j=1,2,3} |f^{(j)}|_{L^{infty}(0,T_0)}$, where the infimum is taken over every function $f$ which runs through the set $KC^3(0,T_0)$ consisting of all functions $f : [0,T_0] o mathbb{R}$ satisfying the boundary conditions $f^{(j)}(0)=a_j$, $f^{(j)}(T_0)=0$ for $j=0,1,2$, whose derivatives $f^{(j)}$ are continuous for $j=0,1,2$ and the third derivative $f^{(3)}$ may have a finite number of discontinuities in the interval $(0,T_0)$, and find this infimum explicitly for certain choice of boundary conditions. This problem is motivated by some conditions under which the solution of the nonlinear Schr?dinger equation with periodic boundary condition blows up in a finite time.