The fractional Hardy inequality with a remainder term

We calculate the regional fractional Laplacian on some power function on an interval. As an application, we prove Hardy inequality with an extra term for the fractional Laplacian on the interval with the optimal constant. As a result, we obtain the fractional Hardy inequality with best constant and an extra lower-order term for general domains, following the method developed by M. Loss and C. Sloane [arXiv:0907.3054v1 [math.AP]]Comment: Major change

Time-dependent Radiation-driven Winds

We study temporal variability of radiation-driven winds using one-dimensional, time-dependent simulations and an extension of the classic theory of line-driven winds developed by Castor Abbott & Klein. We drive the wind with a sinusoidally varying radiation field and find that after a relaxation time, determined by the propagation time for waves to move out of the acceleration zone of the wind, the solution settles into a periodic state. Winds driven at frequencies much higher than the dynamical frequency behave like stationary winds with time averaged radiation flux, whereas winds driven at much lower frequencies oscillate between the high and low flux stationary states. Most interestingly, we find a resonance frequency near the dynamical frequency that results in velocity being enhanced or suppressed by a factor comparable to the amplitude of the flux variation. Whether the velocity is enhanced or suppressed depends on the relative phase between the radiation and the dynamical variables. These results suggest that a time-varying radiation source can induce density and velocity perturbations in the acceleration zones of line-driven winds

Hardy inequalities and non-explosion results for semigroups

We prove non-explosion results for Schr\"odinger perturbations of symmetric transition densities and Hardy inequalities for their quadratic forms by using explicit supermedian functions of their semigroups.Comment: 21 pages, updated reference