Sharp bounds for Dirichlet eigenvalue ratios of the Camassa-Holm equations

In this paper, we present a short proof of the maximization of Dirichlet eigenvalue ratios for the Camassa-Holm equation y " = 1/4 y + lambda m(x)y, by solving the infinitely dimensional maximization problem R-k(r, B) = sup(m is an element of E(r,B)) lambda(2k)(m)/lambda(1)(m) , k is an element of N, when the potentials satisfy that ||m||(1) <= r and m(x) <= -B for some constants r > 0 and B is an element of (0, r]. The maximization will be given as an elementary function. Our results shed new lights on such kind of problems because we do not require the potentials to be symmetric or monotone. Because the solution of the maximization problem leads to more general distributions of potentials which have no densities with respect to the Lebesgue measure, we choose the general setting of the measure differential equations dy(center dot) = 1/4 ydx + yd mu(x), to understand such problems.