On a Shape Derivative Formula with Respect to Convex Domains

We extend to the case of W ^1,1 _loc functions a formula, known for C ¹ functions, for the computation of the shape derivative of an integral cost functional with respect to a class of admissible convex domains. The convexity context allows one to use the support functions of the domains to express the shape derivative. We shall also illustrate the formula by applying it to the computation of the shape derivative for a shape optimization problem and by giving an algorithm