OPTIMAL SHAPES FOR TREE ROOTS

Abstract. The paper studies a class of variational problems, modeling optimal shapes for tree roots. Given a measure μ describing the distribution of root hair cells, we seek to maximize a harvest functional H, computing the total amount of water and nutrients gathered by the roots subject to a cost for transporting these nutrients from the roots to the trunk. Earlier papers have established the existence of an optimal measure and a priori bounds. Here we derive necessary conditions for optimality. Moreover, in space dimension d = 2, we prove that the support of an optimal measure is nowhere dense.