Some Variational Problems Involving Nonlocal Perimeters and Applications

The thesis is on several minimization problems involving nonlocal perimeters. The nonlocal perimeter is a nonlocal extension of the classical perimeter. The thesis is written on my contributions with several collaborators and these contributions can be found in the articles in [49, 96, 97, 99]. We have mainly investigated three problems: nonlocal minimal surfaces, nonlocal denoising problems, and nonlocal liquid drop models. After giving a brief introduction on the nonlocal perimeters and its motivation in Chapter 1, we give the definition of nonlocal(fractional) perimeters in Chapter 2. Moreover, in Chapter 2, we collect several properties of the nonlocal(fractional) perimeters and also give some of their proofs. We also mention the regularity of sets which are (almost) minimizers of the nonlocal(fractional) perimeter. In Chapter 3, we study the topology of nonlocal(fractional) minimal surfaces in a specific situation. The study of the nonlocal(fractional) minimal surfaces has been initiated by L. Caffarelli, J.M. Roquejoffre, and O. Savin [22] and, since then, this topic has attracted many authors. In particular, S. Dipierro, O. Savin, and E. Valdinoci [50, 51, 52] discovered the “stickiness” property of the nonlocal minimal surfaces, which is not true in the case of the classical minimal surfaces. Motivated by these works, we study how the shape of the nonlocal minimal surfaces look like in a specific cylinder with an initial data given by the complement of a slab perpendicular to the cylinder. In this setting, we prove that, if the width of the slab is small enough, then the nonlocal minimal surfaces coincide with the cylinder and, if the width is large enough, then the nonlocal minimal surfaces tend to stick to the boundary of the cylinder. The first result implies that nonlocal minimal surfaces cannot develop catenoids in some situation. This is not the case in the classical minimal surfaces. In Chapter 4, we consider a nonlocal extension of the denoising model which was introduced by L. Rudin, S. Osher, and E. Fatemi [104]. Our denoising model is formulated as the minimization problem of the energy consisting of the nonlocal(fractional) total variation and L2-fidelity term. The denoising model can be applied to remove noises from given images and recover the original images. In this thesis, we are particularly interested in the regularity of the (unique) minimizer of the energy. We obtain that, in 2 dimension, the minimizer is as regular as the given data (of class C0,α). This result can be regarded as a nonlocal version of the result by V. Caselles, A. Chambolle, and M. Novaga [30]. In Chapter 5, we consider a nonlocal extension of the liquid drop model which was introduced by G. Gamow [63] in 1930s. Our model is formulated as the minimization problem, with volume constraint, of the energy consisting of the nonlocal(fractional) perimeter and generalized Riesz potential term. The classical model was studied in order to explain the behaviour of atomic nuclei and predict nuclear fission. Heuristically, one can see that, if the volume is large, then the Riesz term dominates the perimeter term and, if the volume is small, then the perimeter term dominates the Riesz term. The former implies the nonexistence of minimizers (nuclear fission) and the latter implies the existence of minimizers (stability of atomic nuclei). In the classical case, there are a lot of works on the model [71, 72, 69, 83, 12, 100, 93] (not exhausted); however, the nonlocal case is not well-understood (see [56, 27] for small mass regime). In this thesis, we are interested in the minimizers for large volumes. We obtain that, if the kernel of the Riesz term decays much faster than that of the nonlocal perimeter, then there exists a minimizer for any volume. On the other hand, if the kernel of the Riesz term is “properly” controlled by that of the nonlocal perimeter, then there exists no minimizer for large volumes. Moreover, if the Riesz term strongly dominates the perimeter term, then each minimizer converges to a ball as the volume diverges. In Appendix A and B, we give several properties of the nonlocal(fractional) perimeter. In Appendix A, we state the compactness of sets of finite nonlocal perimeters with a general kernel. The proof is based on the results by E. Di Nezza, G. Palatucci, and E. Valdinoci [46]. In Appendix B, we show the Euler-Lagrange equations for minimizers of our functional studied in Chapter 4. The proof is based on the results by M.C. Caputo and N. Guillen [26].