2018/11/19 by Annalisa Cesaroni, Serena Dipierro, Cesaroni, Annalisa +5
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1811.07621
openalex publication_date 2018/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider an energy functional combining the square of the local\noscillation of a one--dimensional function with a double well potential. We\nestablish the existence of minimal heteroclinic solutions connecting the two\nwells of the potential.\n This existence result cannot be accomplished by standard methods, due to the\nlack of compactness properties.\n In addition, we investigate the main properties of these heteroclinic\nconnections. We show that these minimizers are monotone, and therefore they\nsatisfy a suitable Euler-Lagrange equation.\n We also prove that, differently from the classical cases arising in ordinary\ndifferential equations, in this context the heteroclinic connections are not\nnecessarily smooth, and not even continuous (in fact, they can be piecewise\nconstant). Also, we show that heteroclinics are not necessarily unique up to a\ntranslation, which is also in contrast with the classical setting.\n Furthermore, we investigate the associated Dirichlet problem, studying\nexistence, uniqueness and partial regularity properties, providing explicit\nsolutions in terms of the external data and of the forcing source, and\nexhibiting an example of discontinuous solution.\n