2023/10/20 by Matías G. Delgadino, Michael Vaughan, Delgadino, Matias G. +1 · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Fractional Differential Equations Solutions
paper · pdf · doi:10.48550/arxiv.2310.13221
We show that nonlocal seminorms are strictly decreasing under the continuous Steiner rearrangement. This implies that all solutions to nonlocal equations which arise as critical points of nonlocal energies are radially symmetric and decreasing. Moreover, we show uniqueness of solutions by exploiting the convexity of the energies under a tailored interpolation in the space of radially symmetric and decreasing functions. As an application, we consider the long time dynamics of a higher order nonlocal equation which models the growth of symmetric cracks in an elastic medium.