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

We develop a variational approach to the minimization problem of functionals of the type $\frac12\left\lVert \nabla \phi \right\rVert^2_2 + \beta \left\lVert \phi \right\rVert_1$ constrained by $\left\lVert \phi \right\rVert_2 = 1$ 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 $\beta$.