Margherita Solci

Variational models of phase transitions take into account double-well energies singularly perturbed by gradient terms, such as the Cahn–Hilliard free energy. The derivation by [Formula: see text]-convergence of a sharp-interface limit for such energies is a classical result by Modica and Mortola. We consider a singular perturbation of a double-well energy by derivatives of order [Formula: see text] and show that we still can describe the limit as in the case [Formula: see text] with a suitable interfacial energy density, in accordance with the case [Formula: see text] and with the case [Formula: see text] previously analyzed by Fonseca and Mantegazza. The main issue is the derivation of an optimal-profile problem on the real line describing the interfacial energy density, which must be conveniently approximated by minimum problems on finite intervals with homogeneous condition on the derivatives at the endpoints up to order [Formula: see text]. To that end, a careful study of sets where sequences of functions with equibounded energy are “close to the wells” and have “small derivatives” in terms of interpolation inequalities and energy estimates must be carried out.

We consider sequences of quadratic nonlocal functionals, depending on a small parameter [Formula: see text], that approximate the Dirichlet integral by a well-known result by Bourgain, Brezis, and Mironescu. Similarly to what is done for core-radius approximations to vortex energies in the case of the Dirichlet integral, we further scale such energies by [Formula: see text] and restrict them to [Formula: see text]-valued functions. We introduce a notion of convergence of functions to integral currents with respect to which such energies are equicoercive, and show the convergence to a vortex energy, similarly to the limit behavior of Ginzburg–Landau energies at the vortex scaling.

We study a discrete approximation of functionals depending on nonlocal gradients. The discretized functionals are proved to be coercive in classical Sobolev spaces. The key ingredient in the proof is a formulation in terms of circulant Toeplitz matrices.

We consider energies on a periodic set ℒ {\mathcal{L}} of the form ∑ i , j ∈ ℒ a i ⁢ j ε ⁢ | u i - u j | {\sum_{i,j\in\mathcal{L}}a^{\varepsilon}_{ij}\lvert u_{i}-u_{j}\rvert} , defined on spin functions u i ∈ { 0 , 1 } {u_{i}\in\{0,1\}} , and we suppose that the typical range of the interactions is R ε {R_{\varepsilon}} with R ε → + ∞ {R_{\varepsilon}\to+\infty} , i.e., if | i - j | ≤ R ε {\lvert i-j\rvert\leq R_{\varepsilon}} , then a i ⁢ j ε ≥ c > 0 {a^{\varepsilon}_{ij}\geq c>0} . In a discrete-to-continuum analysis, we prove that the overall behavior as ε → 0 {\varepsilon\to 0} of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on ε ⁢ ℒ {\varepsilon\mathcal{L}} with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded R ε {R_{\varepsilon}} and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case ℒ = ℤ d {\mathcal{L}=\mathbb{Z}^{d}} .