Shift Differentials of Maps in BV Spaces. Alberto Bressan and Marta Lewicka SISSA Ref. 59/98/M (June, 1998) 1 Introduction Aim of this note is to provide a brief outline of the theory of shift-differentials, introduced in [2], and show how their construction can be extended to the case of vector valued functions. In the following, we consider the space BV of scalar integrable functions having bounded variation, endowed with the L 1 norm. We recall that, given a map Φ : X → Y between normed linear spaces, its differential at a point x 0 is the linear map Λ: X → Y such that lim h→0 ‖Φ(x 0 + h) − Φ(x 0 ) − Λ(h)‖ Y ‖h‖ X =0. (1.1) This concept of differential (see for example [6]) is one of the cornerstone of math- ematical analysis, providing a basic tool in the study of regular maps. For maps which do not admit a first-order linear approximation, various concepts of weak or generalized differential can be found in the literature [4] [7] [9] [11]. The present paper intends to provide some further contribution in this direction. The primary motivation for the introduction of shift differentials comes from the the- ory of hyperbolic conservation laws [8] [10]. As a simple example, consider Burgers’ equation u t +[u 2 /2] x =0 (1.2) with the family of initial conditions u θ (0,x)= θx · χ [0,1] (x). (1.3) 1