On equality of the $L^\infty$ norm of the gradient under the Hausdorff and Lebesgue measure

TL;DR

This paper proves that the L∞-norm of the gradient is invariant when passing from k-dimensional Hausdorff measure to n-dimensional Lebesgue measure for functions differentiable almost everywhere with respect to the k-dimensional measure. The proof proceeds via a staged strategy: a one-dimensional base case, extension to higher dimensions through Fubini and Lipschitz arguments, and a final slicing argument along spheres to transfer gradient bounds to almost every hypersurface. The main contributions are the gradient-norm equality, the preservation of everywhere differentiability under W^{1,∞} convergence, and the closedness of C^1(Ω) in W^{1,∞}(Ω), with implications for regularity in the L∞ calculus of variations. The work provides tools to analyze limits of Lipschitz and infinity-harmonic-type sequences in PDE and geometric measure theory.

Abstract

Let $Ω$ be an open subset of $\mathbb R^n$, and let $f: Ω\to \mathbb R$ be differentiable $\mathcal H^k$-almost everywhere, for some nonnegative integer $k < n$, where $\mathcal H^k$ denotes the $k$=dimensional Hausdorff measure. We show that $\|\nabla f\|_{L^\infty (\mathcal H^k)} = \|\nabla f\|_{L^\infty(\mathcal H^n)}.$ We deduce that convergence in the Sobolev space $W^{1, \infty}$ preserves everywhere differentiability. As a further corollary, we deduce that the class $C^1 (Ω)$ of continuously differentiable functions is closed in $W^{1, \infty}(Ω)$.

Theorems & Definitions (11)

• Theorem 1
• Proposition 2
• Corollary 3
• Proposition 4
• proof
• Proposition 5
• proof
• proof : Proof of Theorem 1
• Lemma 6
• proof : Proof of Proposition \ref{['corollary2']}