A quantitative version of the transversality theorem

TL;DR

This paper develops a quantitative estimate for the measure of the set where a continuous function intersects a smooth manifold, with applications to counting shock curves in scalar conservation laws.

Contribution

It introduces a quantitative version of the transversality theorem, providing measure estimates and applications to scalar conservation laws.

Findings

Established a Hausdorff measure estimate for intersection sets.

Applied the estimate to quantify shock curves in conservation laws.

Provided a new quantitative framework for transversality in analysis.

Abstract

The present paper studies a quantitative version of the transversality theorem. More precisely, given a continuous function $g\in \mathcal{C}([0,1]^d,\mathbb{R}^m)$ and a global smooth manifold $W\subset \mathbb{R}^m$ of dimension $p$, we establish a quantitative estimate on the $(d+p-m)$-dimensional Hausdorff measure of the set $\mathcal{Z}_{W}^{g}=\left\{x\in [0,1]^d: g(x)\in W\right\}$. The obtained result is applied to quantify the total number of shock curves in weak entropy solutions to scalar conservation laws with uniformly convex fluxes in one space dimension.