arXiv subjects

Matteo Cozzi

Publications and source records attributed to Matteo Cozzi.

25 records · Page 2Linked to original sources

arXiv2018

Planelike minimizers of nonlocal Ginzburg-Landau energies and fractional perimeters in periodic media

We consider here a nonlocal phase transition energy in a periodic medium and we construct solutions whose interfaces lie at a bounded distance from any given hyperplane. These solutions are either periodic or quasiperiodic, depending on the rational dependency of the normal direction to the reference hyperplane. Remarkably, the oscillations of the interfaces with respect to the reference hyperplane are bounded by a universal constant times the periodicity scale of the medium. This geometric property allows us to establish, in the limit, the existence of planelike nonlocal minimal surfaces in a periodic structure. The proofs rely on new optimal density and energy estimates. In particular, roughly speaking, the energy of phase transition minimizers is controlled, both from above and below, by the energy of one-dimensional transition layers.

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arXiv2016

Planelike interfaces in long-range Ising models and connections with nonlocal minimal surfaces

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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arXiv2016

Nonlocal phase transitions in homogeneous and periodic media

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 $Γ$-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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arXiv2016

Interior regularity of solutions of non-local equations in Sobolev and Nikol'skii spaces

We prove interior $H^{2s-\varepsilon}$ regularity for weak solutions of linear elliptic integro-differential equations close to the fractional $s$-Laplacian. The result is obtained via intermediate estimates in Nikol'skii spaces, which are in turn carried out by means of an appropriate modification of the classical translation method by Nirenberg.

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arXiv2015

One-dimensional solutions of non-local Allen-Cahn-type equations with rough kernels

We are interested in the study of local and global minimizers for an energy functional of the type $$ \frac{1}{4} \iint_{\mathbb{R}^{2 N} \setminus \left( \mathbb{R}^N \setminus Ω\right)^2} |u(x) - u(y)|^2 K(x - y) \, dx dy + \int_Ω W(u(x)) \, dx, $$ where $W$ is a smooth, even double-well potential and $K$ is a non-negative symmetric kernel in a general class, which contains as a particular case the choice $K(z) = |z|^{- N - 2 s}$, with $s \in (0, 1)$, related to the fractional Laplacian. We show the existence and uniqueness (up to translations) of one-dimensional minimizers in the full space $\mathbb{R}^N$ and obtain sharp estimates for some quantities associated to it. In particular, we deduce the existence of solutions of the non-local Allen-Cahn equation $$ \mbox{p.v.} \int_{\mathbb{R}^N} \left( u(x) - u(y) \right) K(x - y) \, dy + W'(u(x)) = 0 \quad \mbox{for any } x \in \mathbb{R}^N, $$ which possess one-dimensional symmetry. The results presented here were proved in (Cabré and Solà-Morales, 2005), (Palatucci, Savin and Valdinoci, 2013) and (Cabré and Sire, 2015) for the model case $K(z) = |z|^{- N - 2 s}$. In our work, we consider instead general kernels which may be possibly non-homogeneous and truncated at infinity.

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arXiv2014

Gradient bounds and rigidity results for singular, degenerate, anisotropic partial differential equations

We consider the Wulff-type energy functional $$ \mathcal{W}_Ω(u) := \int_ΩB(H(\nabla u (x))) - F(u(x)) \, dx, $$ where $B$ is positive, monotone and convex, and $H$ is positive homogeneous of degree 1. The critical points of this functional satisfy a possibly singular or degenerate, quasilinear equation in an anisotropic medium. We prove that the gradient of the solution is bounded at any point by the potential $F(u)$ and we deduce several rigidity and symmetry properties.

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arXiv2014

Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs

We consider possibly degenerate and singular elliptic equations in a possibly anisotropic medium. We obtain monotonicity results for the energy density, rigidity results for the solutions and classification results for the singularity/degeneracy/anisotropy allowed. As far as we know, these results are new even in the case of non-singular and non-degenerate anisotropic equations.

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