Related papers: Fractional multi-phase transitions and nonlocal mi…
In this paper, we are concerned with the asymptotic behavior of weak solutions to certain elliptic and parabolic problems involving the fractional $p$-Laplacian in cylindrical domains that become unbounded in one direction. The nonlocal…
We consider a model of energy minimization arising in the study of the mechanical behavior caused by cell contraction within a fibrous biological medium. The macroscopic model is based on the theory of non rank-one convex nonlinear…
In this paper, we propose and analyze a time-stepping method for the time fractional Allen-Cahn equation. The key property of the proposed method is its unconditional stability for general meshes, including the graded mesh commonly used for…
This contribution is devoted to cover some technical aspects related to the use of the recently proposed energy probability distribution zeros in the study of phase transitions. This method is based on the partial knowledge of the partition…
The numerical integration of phase-field equations is a delicate task which needs to recover at the discrete level intrinsic properties of the solution such as energy dissipation and maximum principle. Although the theory of energy…
In the first part of this article we deal with the existence of at least three non-trivial weak solutions of a nonlocal problem with nonstandard growth involving a nonlocal Robin type boundary condition. The second part of the article is…
We study entire minimizers of the Allen-Cahn systems. The specific feature of our systems are potentials having a finite number of global minima, with sub-quadratic behaviour locally near their minima. The corresponding formal…
In this short note we present new results on a higher-dimensional generalization of De~Giorgi's conjecture for Allen--Cahn type equations, focusing on dimensions $n \ge 9$. Although counterexamples are known in this regime, our goal is to…
Fractional diffusion has become a fundamental tool for the modeling of multiscale and heterogeneous phenomena. However, due to its nonlocal nature, its accurate numerical approximation is delicate. We survey our research program on the…
Phase-field models of fracture introduce smeared cracks of width commensurate with a regularisation length parameter $\epsilon$ and obeying a minimum energy principle. Mesh adaptivity naturally suggests itself as a means of supplying…
We study a Fokker-Planck equation with double-well potential that is nonlocally driven by a dynamical constraint and involves two small parameters. Relying on formal asymptotics we identify several parameter regimes and derive reduced…
It is well known that, if the initial conditions have sufficiently high energy density, the dynamics of the classical Discrete Non-Linear Schr\"odinger Equation (DNLSE) on a lattice shows a form of breaking of ergodicity, with a finite…
We present for the first time to the nuclear physics community the Hamiltonian Mean Field (HMF) model. The model can be solved analytically in the canonical ensemble and shows a second-order phase transition in the thermodynamic limit.…
The notion of fractional charges was up until now reserved for quasiparticle excitations emerging in strongly correlated quantum systems, such as Laughlin states in the fractional quantum Hall effect, Luttinger quasiparticles, or…
In this work we introduce a new system of partial differential equations as a simplified model for the evolution of reversible martensitic transformations under thermal cycling in low hysteresis alloys. The model is developed in the context…
Integral-type nonlocal damage models describe the fracture process zones by regular strain profiles insensitive to the size of finite elements, which is achieved by incorporating weighted spatial averages of certain state variables into the…
We study a quadratic nonlocal variational problem on a hybrid domain formed by a compact interval and finitely many discrete points. The associated energy splits into continuous, discrete, and interface contributions. Our main estimate…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…
A novel method is developed for constructing periodic solutions of a model equation describing nonlocal Josephson electrodynamics. This method consists of reducing the equation to a system of linear ordinary differential equations through a…
Nuclear matter at finite temperature and barion density exhibits several phase transitions that could happen at the early stages of the Universe evolution and could be realized in heavy-ion or hadron-hadron collisions. Microscopic…