On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4+1

We prove that every bounded stable solution of (-Delta)1/2u+f(u)=0 in R3 is a 1D profile, i.e., u(x)=phi(e center dot x) for some e is an element of S2, where phi:R -> R is a nondecreasing bounded stable solution in dimension one. Equivalently, stable critical points of boundary reaction problems in R+d+1=Rd+1 boolean AND{xd+1 >= 0} of the form {xd+1 >= 0}12| are 1D when d=3 These equations have been studied since the 1940's in crystal dislocations. Also, as it happens for the Allen-Cahn equation, the associated energies enjoy a Gamma convergence result to the perimeter functional. In particular, when f(u)=u3-u, our result implies the analogue of the De Giorgi conjecture for the half-Laplacian in dimension 4, namely that monotone solutions are 1D. Note that our result is a PDE version of the fact that stable embedded minimal surfaces in R3 are planes. It is interesting to observe that the corresponding statement about stable solutions to the Allen-Cahn equation (namely, when the half-Laplacian is replaced by the classical Laplacian) is still unknown for d=3.