Given a smooth u : R → R, say u = u(y), we consider u = u(x, y) to be a solution of ∆u = 0 for any (x, y) ∈ (0, 1)× R, u(0, y) = u(y) for any y ∈ R, ux(1, y) = 0 for any y ∈ R. We define the Dirichlet-Neumann operator (L u)(y) = ux(0, y) and we prove a symmetry result for equations of the form (L u)(y) = f(u(y)). In particular, bounded, monotone solutions in R are proven to depend only on one Euclidean variable. Introduction The aim of this paper is to provide a symmetry result for a Dirichlet-Neumann problem. Our set up is the following. We consider the slab [0, 1] × R, endowed with coordinates x ∈ [0, 1] and y ∈ R. We define the operator L as follows. Given a smooth u, which will be taken to be bounded together with its derivatives, we define u(x, y) ∈ C((0, 1)×R)∩ C([0, 1]× R) to be the solution of (1) ∆u = 0 in (0, 1)× R, u(0, y) = u(y), ux(1, y) = 0. As customary, the subscript denotes the partial derivative and ∆u = uxx + uy1y1 + · · · + uynyn is the Laplace operator. The problem in (1) is well-posed and it possesses nice regularity properties, due to the elliptic PDE theory (see, e.g., Theorems 6.6 and 6.26 in [GT01]). Then, we define (L u)(y) = ux(0, y). The linear operator L may also be written in the harmonic analysis setting. That is, if F denotes the Fourier transform in the y variables (and the transformed 1 2 RAFAEL DE LA LLAVE AND ENRICO VALDINOCI frequency variables are called ξ ∈ R), we have that (2) L u = F−1 ( |ξ| −|ξ| − e|ξ| e−|ξ| + e|ξ| (Fu)(ξ) ) , up to a normalization factor. From (2), we may say that the symbol of the operator L in Fourier space is (3) |ξ| −|ξ| − e|ξ| e−|ξ| + e|ξ| . Though Fourier analysis will not explicitly play much of a role in this paper, it is convenient to keep in mind that, for large frequencies ξ, (3) is asymptotic to the symbol of the square root of the Laplacian. The operator L arises in the theory of water waves of irrotational, incompressible, inviscid fluids in the small amplitude, long wave regime [Sto57, Zak68, Whi74, CSS92, CG94, NS94, CW95, dlLP96, CSS97, CN00, GG03, HN05, NT08]. Related nonlocal operators are studied in flame propagation and semipermeable membranes [CRS07], in optimization [DL76], in relation with the ultrarelativistic limit of quantum mechanics [FdlL86], in the theory of quasi-geostrophic flows [MT96, Cor98] in inverse spectral and multiple scattering problems [DG75, CK98, GK04] and in the thin obstacle problem [Caf79]. Of course, these operators are also a classical topic in harmonic analysis and in singular integral theory [Lan72, Ste70]. The main result that we prove is the following: Theorem 1. Let f ∈ C(R). Let u be a bounded solution of (L u)(y) = f(u(y)) for any y ∈ R. Suppose that (4) uyn(y) > 0 for any y ∈ R and that there exists C > 0 such that