Sets Of Finite Perimeter And Geometric Variational Problems PDF Download

This thesis is dedicated to the study of two separate geometric variational problems involving nonlocal energies: firstly, the geometry and singularities of fractional harmonic maps,and secondly, an iso perimetric problem with a repulsive integrable potential inspired by Gamow's liquid drop model for the atomic nucleus. On the first topic, we improve already-known results for minimizing 1/2-harmonic maps when the target manifold is a sphere by reducing the upperbound on the Haudorff dimension of the singular set, i.e., the set of points of discontinuity. Wealso characterize so-called minimizing 1/2-harmonic tangent maps from the plane into the unit circle S1, shedding light on the behavior of minimizing 1/2-harmonic maps from R2into S1 near singularities. Finally, when s ∈ (0, 1), we prove partial regularity results for s-harmonic maps into spheres in the stationary and minimizing case, obtaining sharp estimates on the Hausdorffd imension of the set of singularities, depending on the value of s. As for the second topic of the thesis, we study a minimization problem on sets of finite perimeter under a volume constraint, where the functional is the sum of a cohesive perimeter term and a repulsive term given by a general integrable symmetric kernel on Rn. We show that under reasonable assumptions on the behavior near the origin and on some of the moments of this kernel - which include physically relevant Bessel potentials - the problem admits large mass (or volume) minimizers. In addition,after normalization, those minimizers converge to the unit ball as the mass goes to infinity. By studying the stability of the ball, we show that without these assumptions, symmetry breaking can occur, that is, there are cases when the problem admits minimizers which cannot be the ball.