School of Mathematics and Statistics - Research Publications
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ItemAsymptotic Expansions of the Contact Angle in Nonlocal Capillarity Problems(SPRINGER, 2017-10)
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ItemNonlocal phase transitions in homogeneous and periodic media(SPRINGER BASEL AG, 2017-03)We discuss some results related to a phase transition model in which the potential energy induced by a double-well function is balanced by a fractional elastic energy. In particular, we present asymptotic results (such as $\Gamma$-convergence, energy bounds and density estimates for level sets), flatness and rigidity results, and the construction of planelike minimizers in periodic media. Finally, we consider a nonlocal equation with a multiwell potential, motivated by models arising in crystal dislocations, and we construct orbits exhibiting symbolic dynamics, inspired by some classical results by Paul Rabinowitz.
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ItemOn fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results(Springer, 2017-12-01)We consider a nonlocal equation set in an unbounded domain with the epigraph property. We prove symmetry, monotonicity and rigidity results. In particular, we deal with halfspaces, coercive epigraphs and epigraphs that are flat at infinity. These results can be seen as the nonlocal counterpart of the celebrated article (Berestycki et al., Commun Pure Appl Math 50(11):1089–1111, 1997).
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ItemPlanelike Interfaces in Long-Range Ising Models and Connections with Nonlocal Minimal Surfaces(SPRINGER, 2017-06)This paper contains three types of results: 1. the construction of ground state solutions for a long-range Ising model whose interfaces stay at a bounded distance from any given hyperplane, 2. the construction of nonlocal minimal surfaces which stay at a bounded distance from any given hyperplane, 3. the reciprocal approximation of ground states for long-range Ising models and nonlocal minimal surfaces. In particular, we establish the existence of ground state solutions for long-range Ising models with planelike interfaces, which possess scale invariant properties with respect to the periodicity size of the environment. The range of interaction of the Hamiltonian is not necessarily assumed to be finite and also polynomial tails are taken into account (i.e. particles can interact even if they are very far apart the one from the other). In addition, we provide a rigorous bridge between the theory of long-range Ising models and that of nonlocal minimal surfaces, via some precise limit result.
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ItemNonlocal Diffusion and Applications(SPRINGER, 2016)We consider the fractional Laplace framework and provide models and theorems related to nonlocal diffusion phenomena. Some applications are presented, including: a simple probabilistic interpretation, water waves, crystal dislocations, nonlocal phase transitions, nonlocal minimal surfaces and Schr\"{o}dinger equations. Furthermore, an example of an $s$-harmonic function, the harmonic extension and some insight on a fractional version of a classical conjecture formulated by De Giorgi are presented. Although this book aims at gathering some introductory material on the applications of the fractional Laplacian, some proofs and results are original. Also, the work is self contained, and the reader is invited to consult the rich bibliography for further details, whenever a subject is of interest.