Calculus of Variations and Geometric Measure Theory

A. Marchese - A. Schioppa

Lipschitz functions with prescribed blowups at many points

created by marchese on 15 Dec 2016
modified on 29 Jun 2019


Published Paper

Inserted: 15 dec 2016
Last Updated: 29 jun 2019

Journal: Calc. Var. Partial Differential Equations
Year: 2016


In this paper we prove generalizations of Lusin-type theorems for gradients due to Giovanni Alberti, where we replace the Lebesgue measure with any Radon measure $\mu$. We apply this to go beyond the known result on the existence of Lipschitz functions which are non-differentiable at $\mu$-almost every point $x$ in any direction which is not contained in the decomposability bundle $V(\mu,x)$, recently introduced by Alberti and the first named author. More precisely, we prove that it is possible to construct a Lipschitz function which attains any prescribed admissible blowup at every point except for a closed set of points of arbitrarily small measure. Here a function is an admissible blowup at a point $x$ if it is null at the origin and it is the sum of a linear function on $V(\mu,x)$ and a Lipschitz function on $V(\mu,x)^{\perp}$.

Keywords: Lusin type approximation, Lipschitz function, Radon measure, Blowup