Abstract
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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Notes
We stress that the case in which \(s \in (0, 1/2]\) can be treated afterwards via a limiting argument, approximating K with kernels truncated at infinity. The difficulties arise essentially for the fact that when \(s \in (0,1/2]\), the functional \(\mathcal {F}\) is identically equal to \(+\infty \) on \({\mathcal {A}}_\omega ^M\), due to the “fat” tails of the kernel.
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Acknowledgements
This work has been supported by the ERC Grant 277749 “EPSILON Elliptic PDE’s and Symmetry of Interfaces and Layers for Odd Nonlinearities”, the PRIN Grant 201274FYK7 “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations” and the Alexander von Humboldt Foundation.
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In honor of Professor Paul Rabinowitz, with great esteem and admiration.
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Cozzi, M., Dipierro, S. & Valdinoci, E. Nonlocal phase transitions in homogeneous and periodic media. J. Fixed Point Theory Appl. 19, 387–405 (2017). https://doi.org/10.1007/s11784-016-0359-z
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DOI: https://doi.org/10.1007/s11784-016-0359-z