LAYER SOLUTIONS IN A HALFSPACE FOR BOUNDARY REACTIONS XAVIER CABR ´ E AND JOAN SOL ` A-MORALES 1. Introduction This article is concerned with the nonlinear problem    Δu =0 in R n + ∂u ∂ν = f (u) on ∂ R n + , (1.1) where n ≥ 2, R n + = {(x,y) ∈ R × R n−1 : x> 0} is a halfspace, ∂ R n + = {x =0}, u is real valued, and ∂u/∂ν = −u x is the exterior normal derivative of u. Points in R n−1 are denoted by y =(y 1 ,...,y n−1 ). Our main goal is to study bounded solutions of (1.1) which 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 (u) = sin(cu) for some constant c, problem (1.1) in a halfplane is called the Peierls-Nabarro problem and it appears as a model of dislocations in crystals (see [33, 20]). The Peierls-Nabarro problem is also central for the analysis of boundary vortices in the paper [26], which studies a model for soft thin films in micromagnetism recently derived by Kohn and Slastikov [24] (see also [25]). 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 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, that 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 two (see Theorem 1.3). The estimate can be seen as an analogue of the pointwise Modica estimate [28] for entire solutions of equations with reaction in the interior. Another important tool throughout the paper consists of establishing relations be- tween layer solutions and two other classes of solutions: local minimizers and stable solutions of (1.1). This is in the spirit of similar ideas carried out for interior reac- tions by Alberti, Ambrosio and one of the authors in [1]. We prove that every layer 1