for very interesting and pleasant discussions. We are also indebted with all the participants of the seminars and minicourses from which this set of notes generated for the nice feedback received, and we hope that this work, though somehow sketchy and informal, can be useful to stimulate new discussions and further develop this rich and interesting subject.

We consider a fractional Laplace equation and we give a selfcontained elementary exposition of the representation formula for the Green function on the ball. In this exposition, only elementary calculus techniques will be used, in particular, no probabilistic methods or computer assisted algebraic manipulations are needed. The main result in itself is not new (see for instance [2,9]), however we believe that the exposition is original and easy to follow, hence we hope that this paper will be accessible to a wide audience of young researchers and graduate students that want to approach the subject, and even to professors that would like to present a complete proof in a PhD or Master Degree course.

Abstract. We prove that the (nonlocal) Marchaud fractional derivative in R can be obtained from a parabolic extension problem with an extra (positive) variable, as the operator that maps the heat conduction equation to the Neumann condition. Some properties of the fractional derivative are deduced from those of the local operator. In particular we prove a Harnack principle for Marchaud-stationary functions.

In this paper, we consider the asymptotic behavior of the fractional mean curvature when s → 0 + . Moreover, we deal with the behavior of s-minimal surfaces when the fractional parameter s ∈ (0, 1) is small, in a bounded and connected open set with C 2 boundary Ω ⊂ R n . We classify the behavior of s-minimal surfaces with respect to the fixed exterior data (i.e. the s-minimal set fixed outside of Ω). So, for s small and depending on the data at infinity, the s-minimal set can be either empty in Ω, fill all Ω, or possibly develop a wildly oscillating boundary.Also, we prove the continuity of the fractional mean curvature in all variables, for s ∈ [0, 1]. Using this, we see that as the parameter s varies, the fractional mean curvature may change sign.