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Quantitative regularity properties for the optimal design problem

Lorenzo Lamberti, Antoine Lemenant

TL;DR

The work advances the regularity theory for the optimal design problem by proving that the boundary of the optimal set is uniformly rectifiable for quasi-minimizers in any dimension, enabling quantitative geometric control. In dimension two, it delivers a full regularity result when the contrast between the Dirichlet coefficients satisfies $β<4α$, using a monotonicity formula and integration-by-parts arguments; it also provides a spectral analysis in two-point contact scenarios and a quantitative separation between components. The combination of uniform rectifiability, a Carleson-measure flatness framework, and 2D monotonicity techniques yields both sharp regularity results and robust structural information (e.g., component distance bounds) that partially address longstanding questions raised by Larsen. The results bridge geometric measure theory with variational regularity for two-phase free boundary problems, with implications for understanding the geometry of optimal designs.

Abstract

In this paper we slightly improve the regularity theory for the so called optimal design problem. We first establish the uniform rectifiability of the boundary of the optimal set, for a larger class of minimizers, in any dimension. As an application, we improve the bound obtained by Larsen in dimension~2 about the mutual distance between two connected components. Finally we also prove that the full regularity in dimension 2 holds true provided that the ratio between the two constants in front of the Dirichlet energy is not larger than 4, which partially answers to a question raised by Larsen.

Quantitative regularity properties for the optimal design problem

TL;DR

The work advances the regularity theory for the optimal design problem by proving that the boundary of the optimal set is uniformly rectifiable for quasi-minimizers in any dimension, enabling quantitative geometric control. In dimension two, it delivers a full regularity result when the contrast between the Dirichlet coefficients satisfies , using a monotonicity formula and integration-by-parts arguments; it also provides a spectral analysis in two-point contact scenarios and a quantitative separation between components. The combination of uniform rectifiability, a Carleson-measure flatness framework, and 2D monotonicity techniques yields both sharp regularity results and robust structural information (e.g., component distance bounds) that partially address longstanding questions raised by Larsen. The results bridge geometric measure theory with variational regularity for two-phase free boundary problems, with implications for understanding the geometry of optimal designs.

Abstract

In this paper we slightly improve the regularity theory for the so called optimal design problem. We first establish the uniform rectifiability of the boundary of the optimal set, for a larger class of minimizers, in any dimension. As an application, we improve the bound obtained by Larsen in dimension~2 about the mutual distance between two connected components. Finally we also prove that the full regularity in dimension 2 holds true provided that the ratio between the two constants in front of the Dirichlet energy is not larger than 4, which partially answers to a question raised by Larsen.

Paper Structure

This paper contains 9 sections, 27 theorems, 225 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminary definitions
  3. Uniform rectifiabilty for quasi-minimizers in dimension $N$
  4. Control of the flatness in a quantitative way
  5. Regularity in dimension $N=2$
  6. Full regularity in the case $\beta < 4\alpha$ from a monotonicity formula
  7. Monotonicity formula for a boundary intersecting by only two points
  8. The components of $A$ have mutually quantitative positive distance
  9. Appendix

Key Result

Theorem 1.1

Let $N=2$ and let $(u,E)$ be a minimizer for the optimal design Problem optimalDesign. Assume moreoever that $\beta< 4 \alpha$. Then $\partial E$ is a smooth $C^{1,\alpha}$-surface in $\Omega$.

Figures (2)

  • Figure 1: The domain noted by $I$ used as a competitor in the proof of Lemma \ref{['LemmaRectangle']}, following the idea of Larsen Larsen2003: if the two connected components $E_1$ and $E_2$ were not flat enough in $B_R$, then we could win a lot of perimeter by adding the domain denoted by $I$ in the picture.
  • Figure 2: In the proof of Theorem \ref{['main2']}, we get a contradiction by finding a ball in which $\partial E$ should be a smooth surface thanks to the $\varepsilon$-regularity result, which prevents a component to be shrinked in a thin rectangle.

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1: Alhfors-regularity
  • Definition 2.2: Uniform rectifiability
  • Definition 2.3: Carleson sets
  • Proposition 2.4
  • Definition 2.5: condition-B
  • Proposition 2.6
  • Theorem 2.7: EspLamPis
  • ...and 46 more