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Carrot John domains in variational problems

Weicong Su, Yi Ru-Ya Zhang

TL;DR

The paper develops a general Minkowski-norm framework for carrot John domains and proves lower semicontinuity of the (optimal) John constant under Hausdorff convergence for bounded John domains, with quantitative bounds and continuity properties of the carrot-John function. In addition, it shows that unbounded open sets satisfying the $J$-carrot John condition with center at infinity can be decomposed into a uniformly finite collection of unbounded $J'$-carrot John subdomains, which jointly cover the relevant region and preserve Sobolev–Poincaré-type inequalities. The results link shape optimization, Teichmüller theory, and Mumford–Shah-type problems by providing stability of John constants under geometric limits and a principled decomposition of unbounded carrot John domains into well-behaved components. These contributions yield robust geometric-analytic tools for variational problems on domains governed by Minkowski norms and carrot/cigar John structures, with potential applications to extremal mappings and regularity theory.

Abstract

In this paper, we explore carrot John domains within variational problems, dividing our examination into two distinct sections. The initial part is dedicated to establishing the lower semicontinuity of the (optimal) John constant concerning Hausdorff convergence for bounded John domains. This result holds promising implications for both shape optimization problems and Techmüller theory. In the subsequent section, we demonstrate that an unbounded open set satisfying the carrot John condition with a center at $\infty$, appearing in the Mumford-Shah problem, can be covered by a uniformly finite number of unbounded John domains (defined conventionally through cigars). These domains, in particular, support Sobolev-Poincaré inequalities.

Carrot John domains in variational problems

TL;DR

The paper develops a general Minkowski-norm framework for carrot John domains and proves lower semicontinuity of the (optimal) John constant under Hausdorff convergence for bounded John domains, with quantitative bounds and continuity properties of the carrot-John function. In addition, it shows that unbounded open sets satisfying the -carrot John condition with center at infinity can be decomposed into a uniformly finite collection of unbounded -carrot John subdomains, which jointly cover the relevant region and preserve Sobolev–Poincaré-type inequalities. The results link shape optimization, Teichmüller theory, and Mumford–Shah-type problems by providing stability of John constants under geometric limits and a principled decomposition of unbounded carrot John domains into well-behaved components. These contributions yield robust geometric-analytic tools for variational problems on domains governed by Minkowski norms and carrot/cigar John structures, with potential applications to extremal mappings and regularity theory.

Abstract

In this paper, we explore carrot John domains within variational problems, dividing our examination into two distinct sections. The initial part is dedicated to establishing the lower semicontinuity of the (optimal) John constant concerning Hausdorff convergence for bounded John domains. This result holds promising implications for both shape optimization problems and Techmüller theory. In the subsequent section, we demonstrate that an unbounded open set satisfying the carrot John condition with a center at , appearing in the Mumford-Shah problem, can be covered by a uniformly finite number of unbounded John domains (defined conventionally through cigars). These domains, in particular, support Sobolev-Poincaré inequalities.
Paper Structure (10 sections, 13 theorems, 206 equations, 5 figures)

This paper contains 10 sections, 13 theorems, 206 equations, 5 figures.

Table of Contents

  1. Introduction
  2. General Minkowski norm.
  3. Bounded John domains
  4. Unbounded open sets satisfying carrot John condition with center $\infty$
  5. Lower-semicontinuity of John constant
  6. John component of unbounded carrot John domain
  7. A decomposition $V_{j,\,R}$ of $B_R\setminus K$
  8. Construction of $\Omega_{j,\,R,\,y}$
  9. Proof of the estimate \ref{['lower es']}
  10. Proof of Lemma \ref{['cigar repla']}

Key Result

Theorem 1.7

Let $J_0\ge 2$ and assume that $\{\Omega_j\}_{j\in\mathbb{N}^+}$ is a sequence of uniformly bounded John domains satisfying for some $c_0> 0$. Then up to passing to a subsequence, $\overline{\Omega}_j$ converges to some compact set $A\subset \mathbb R^n$ in the Hausdorff distance $d_H$ so that the interior $\Omega$ of $A$ satisfies

Figures (5)

  • Figure 1: A domain $\Omega$ is John if, heuristically speaking, it contains a uniformly linearly opened twisted cone at every $x\in \Omega$.
  • Figure 2: The set $K$ is the union of black lines. We apply Bescovitch's covering theorem to cover the set $A_R$ with balls centered at $\partial B_{3R}$.
  • Figure 3: The set $V_{j,\,R}$ may not necessarily be connected. We connect each point in $V_{j,\,R}$ to the curve $\gamma_y[y,y_R]$ using appropriate curves. Subsequently, we take the union of the carrots surrounding these curves to form $\Omega_{j,\,R,\,y}.$
  • Figure 4: The two curves $\gamma_1$ and $\gamma_2$ are presented, respectively, with their end points and the intersection point $y_2$.
  • Figure 5: The set $W_{3,\,R}$ is contained in $W_{2,\,R'}$, and $W_{2,\,R'}$ is contained in $W_{1,\,R"}$, where $R\le R'\le R"$. Eventually they are all contained in $W_{1,\,\infty}$. However, we note that $W_{3,\,R}$ and $W_{3,\,R"}$ could have no intersection.

Theorems & Definitions (34)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: lower-semicontinuity of (optimal) John constants
  • Corollary 1.8
  • proof
  • Theorem 1.9
  • ...and 24 more