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Partial regularity for the optimal $p$-compliance problem with length penalization

Bohdan Bulanyi

TL;DR

The paper analyzes the optimal $p$-compliance problem with a length penalization in a bounded domain, proving that minimizers cannot contain loops and are $C^{1,\alpha}$-regular at $\mathcal{H}^{1}$-a.e. points of the free set inside the domain for all $p\in(N-1,\infty)$. The authors develop a higher-dimensional regularity theory by combining a decay of the $p$-energy under flatness control with an $\varepsilon$-regularity framework, and they introduce a localized energy-difference bound, capacity-based regularity tools, and a density control mechanism to handle the codimension-$N-1$ free boundary. A key technical contribution is the development of barrier constructions and compactness arguments to obtain energy decay and to propagate flatness through scales, culminating in a partial regularity result that extends prior 2D results to any $N\ge2$ and $p> N-1$. The work advances the understanding of free boundary problems with length penalization in the nonlinear $p$-Laplacian setting, with implications for shape optimization and the structure of optimal networks.

Abstract

We establish a partial $C^{1,α}$ regularity result for minimizers of the optimal $p$-compliance problem with length penalization in any spatial dimension $N\geq 2$, extending some of the results obtained in [Chambolle-Lamboley-Lemenant-Stepanov 17], [Bulanyi-Lemenant 20]. The key feature is that the $C^{1,α}$ regularity of minimizers for some free boundary type problem is investigated with a free boundary set of codimension $N-1$. We prove that every optimal set cannot contain closed loops, and it is $C^{1,α}$ regular at $\mathcal{H}^{1}$-a.e. point for every $p\in (N-1,+\infty)$.

Partial regularity for the optimal $p$-compliance problem with length penalization

TL;DR

The paper analyzes the optimal -compliance problem with a length penalization in a bounded domain, proving that minimizers cannot contain loops and are -regular at -a.e. points of the free set inside the domain for all . The authors develop a higher-dimensional regularity theory by combining a decay of the -energy under flatness control with an -regularity framework, and they introduce a localized energy-difference bound, capacity-based regularity tools, and a density control mechanism to handle the codimension- free boundary. A key technical contribution is the development of barrier constructions and compactness arguments to obtain energy decay and to propagate flatness through scales, culminating in a partial regularity result that extends prior 2D results to any and . The work advances the understanding of free boundary problems with length penalization in the nonlinear -Laplacian setting, with implications for shape optimization and the structure of optimal networks.

Abstract

We establish a partial regularity result for minimizers of the optimal -compliance problem with length penalization in any spatial dimension , extending some of the results obtained in [Chambolle-Lamboley-Lemenant-Stepanov 17], [Bulanyi-Lemenant 20]. The key feature is that the regularity of minimizers for some free boundary type problem is investigated with a free boundary set of codimension . We prove that every optimal set cannot contain closed loops, and it is regular at -a.e. point for every .

Paper Structure

This paper contains 14 sections, 37 theorems, 274 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Poincaré inequality
  4. Estimate for $E_{f,\Omega}(u_{\Sigma})-E_{f,\Omega}(u_{\Sigma^{\prime}})$
  5. Uniform boundedness of $u_{f,\Omega,\Sigma}$ with respect to $\Sigma$
  6. Existence
  7. Decay of the $p$-energy under flatness control
  8. Absence of loops
  9. Proof of partial regularity
  10. Control on defect of minimality
  11. Density control
  12. Control of the flatness
  13. Acknowledgments
  14. Auxiliary results

Key Result

Theorem 1.2

Let $\mathop{\mathrm{\Omega}}\limits \subset \mathbb{R}^{N}$ be open and bounded, $p \in (N-1,+\infty),\, f \in L^{q}(\Omega)$ with $q> q_{1}$, where $q_{1}$ is defined in (1.4). Then there exists a constant $\alpha \in (0,1)$ such that the following holds. Let $\mathop{\mathrm{\Sigma}}\limits$ be

Figures (2)

  • Figure 3.1: In the proof of Lemma \ref{['lem 4.1']} we estimate on $\partial\Bigl(\Bigl\{|x^{\prime}|<\frac{\delta_{0}}{\sqrt{2}}\Bigr\}\cap \Bigl\{|x_{N}|<\frac{\delta_{0}}{\sqrt{2}}\Bigr\}\Bigr)$ a nonnegative $p$-harmonic function $u$ in $B_{1}\backslash (\{0\}^{N-1}\times (-1,1))$, continuous in $B_{1}$ with $u=0$ on $\{0\}^{N-1}\times (-1,1)$.
  • Figure 3.2: The geometry in Lemma \ref{['lem 4.3']}.

Theorems & Definitions (83)

  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6
  • proof : Proof of Corollary \ref{['cor 2.6']}
  • Proposition 2.7
  • ...and 73 more