Applications of total positivity theory to 1D prescribed curvature problems

The complexity of global bifurcation diagrams for one-dimensional prescribed curvature problems−(u′/1+|u′|2)′=λf(u),x∈(−L,L),u(−L)=u(L)=0 is closely related to the number of local extreme points of a key function defined by an integral with a parameter. It is known that when the length L passes through the local extrema of the function, the pattern of bifurcation diagrams must change. The more local extreme points the function owns, the more complex the pattern becomes. However, it is very difficult to determine the exact number of local extreme points when the integral function is not monotone over its entire domain. In this paper, we show that total positivity theory of integral operators is a useful tool for estimating this number. By applying this tool to various typical examples, we prove the existence of many new patterns of bifurcation diagrams for one-dimensional prescribed curvature problems. We also make progress on a conjecture of Hung et al. (2014) [12].