A Variational Approach to a $L^1$-Minimization Problem Based on the Milman-Pettis Theorem

Abstract

We develop a variational approach to the minimization problem of functionals of the type 1 2 ∇φ 2 2 + β φ 1 constrained by φ 2 = 1 which is related to the characterization of cases satisfying the sharp Nash inequality. Employing theory of uniform convex spaces by Clarkson and the Milman-Pettis theorem we are able account for the non-reflexivity of L 1 and implement the direct method of calculus of variations. By deriving the Euler-Lagrange equation we verify that the minimizers are up to rearrangement compactly supported solutions to the inhomogeneous Helmholtz equation and we study their scaling behaviour in β.