Layer solutions in a half‐space for boundary reactions

R n−1 are denoted by y = (y1, . . . , yn−1). Our main goal is to study bounded solutions of (1.1) that are monotone increasing, say from −1 to 1, in one of the y-variables. We call them layer solutions of (1.1), and we study their existence, uniqueness, symmetry, and variational properties, as well as their asymptotic behavior. The interest in such increasing solutions comes from some models of boundary phase transitions. When the nonlinearity f is given by f (u) = sin(cu) for some constant c, problem (1.1) in a half-plane is called the Peierls-Nabarro problem, and it appears as a model of dislocations in crystals (see [21, 36]). The Peierls-Nabarro problem is also central to the analysis of boundary vortices in the paper [28], which studies a model for soft thin films in micromagnetism recently derived by Kohn and Slastikov [26] (see also [27]). Our main result, Theorem 1.2, characterizes the nonlinearities f for which there exists a layer solution of (1.1) in dimension n = 2. We prove that the necessary and sufficient condition is that the potential G (defined by G ′ = − f ) has two, and only two, absolute minima in the interval [−1, 1], located at ±1. Under the additional hypothesis G ′′(±1) > 0, we also establish the uniqueness of a layer solution up to translations in the y-variable. The proofs of both the necessity and the sufficiency of the condition on G for existence use new ingredients, which we develop in this article. A first one is a nonlocal estimate, as well as a conserved or Hamiltonian quantity, satisfied by every layer solution in dimension 2 (see Theorem 1.3). The estimate can be seen as