Abstract. We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions.

We consider the optimization problem of minimizing R Ω G(|∇u|) + λχ {u>0} dx in the class of functions W 1,G (Ω) with u − ϕ0 ∈ W 1,G 0 (Ω), for a given ϕ0 ≥ 0 and bounded. W 1,G (Ω) is the class of weakly differentiable functions with R Ω G(|∇u|) dx < ∞. The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω ∩ ∂{u > 0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C 1,α regularity of their free boundaries near "flat" free boundary points.This kind of optimization problem has been widely studied for different functions G. In fact, the first paper in which this problem was studied is [3]. The authors considered the case G(t) = t 2 . They proved that minimizers are weak solutions to the free boundary problem (1.2) ∆u = 0 in {u > 0} u = 0, |∇u| = λ on ∂{u > 0} and proved the Lipschitz regularity of the solutions and the C 1,α regularity of the free boundaries. This free boundary problem appears in several applications. A very important one is that of fluid flow. In that context, the free boundary condition is known as Bernoulli's condition.The results of [3] have been generalized to several cases. For instance, in [5] the authors consider problem (1.1) for a convex function G such that ct < G ′ (t) < Ct for some positive constants c and C. Recently, in the article [7] the authors considered the case G(t) = t p with 1 < p < ∞. In these two papers only minimizers are studied. Minimizers satisfy very good properties like nondegeneracy at the free boundary and uniform positive density of the set {u = 0} at free boundary points. On the other hand, the free boundary problem (1.2) and its

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We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we look at a nonlinear flux boundary condition.

Abstract. We study a singular perturbation problem for a nonlocal evolution operator. The problem appears in the analysis of the propagation of flames in the high activation energy limit, when admitting nonlocal effects.We obtain uniform estimates and we show that, under suitable assumptions, limits are solutions to a free boundary problem in a viscosity sense and in a pointwise sense at regular free boundary points.We study the nonlocal problem both for a single equation and for a system of two equations. Some of the results obtained are new even when the operator under consideration is the heat operator.