Heteroclinic connections for nonlocal equations

Abstract

We construct heteroclinic orbits for a strongly nonlocal integro-differential equation. Since the energy associated to the equation is infinite in such strongly nonlocal regime, the proof, based on variational methods, relies on a renormalized energy functional, exploits a perturbation method of viscosity type, combined with an auxiliary penalization method, and develops a free boundary theory for a double obstacle problem of mixed local and nonlocal type.The description of the stationary positions for the atom dislocation function in a perturbed crystal, as given by the Peierls-Nabarro model, is a particular case of the result presented.

Wellposedness of a nonlinear peridynamic model

“…It is interesting to point out that the functional spaces in which we work allow, in principle, singular functions (see Appendix B in [3]).…”

Lavrentiev gap for some classes of generalized Orlicz functions

“…In general it could happen, that the boundary function has infinite energy J (v). The truncation outside 3Ω allows us to avoid the renormalized energies approach, which was presented in Chen, Muratov, and Yan [CMY17]; Dipierro, Patrizi, and Valdinoci [DPV19].…”

Heteroclinic connections and Dirichlet problems for a nonlocal functional of oscillation type

“…We stress that the existence of heteroclinic connections in nonlocal problems is usually a rather difficult task in itself, which cannot be achieved by standard ordinary differential equations methods and cannot rely directly on conservation of energy formulas. For problems modeled on fractional equations, a careful investigation of heteroclinic solutions and of their basic properties has been recently performed in [2,4,6,14,15,[18][19][20].…”

“…More concretely, we will consider a functional similar to that in (1.2), but in which the derivative is replaced by an oscillation term. We recall that other nonlocal analogues of (1.1) have been considered in the literature, mainly replacing the second derivative with a fractional second derivative, see [4,6,14,19,20]. Other lines of investigation took into account the case in which the second derivative is replaced by a quadratic interaction with an integrable kernel, see [1] and the references therein.…”