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On the L 1-relaxed area of graphs of BV piecewise constant maps taking three values

  • Riccardo Scala EMAIL logo and Giuseppe Scianna
Published/Copyright: October 2, 2024
Advances in Calculus of Variations
From the journal Advances in Calculus of Variations Volume 18 Issue 2

Abstract

Given a bounded open connected set Ω ⊂ ℝ 2 with Lipschitz boundary, we consider the class of piecewise constant maps u taking three fixed values α , β , γ ∈ ℝ 2 , vertices of an equilateral triangle; for any u in this class, using a weak notion of Jacobian determinant valid for BV functions, we give a precise description of Det ⁡ ( ∇ ⁡ u ) and show that the relaxed graph area of u is bounded from above by a quantity related to the flat norm of Det ⁡ ( ∇ ⁡ u ) . The provided upper bound allows to show the validity of a De Giorgi conjecture regarding the relaxed area functional when one restricts to this class of piecewise constant functions.

Keywords: Plateau problem; relaxation; minimal connections; area functional; minimal surfaces; piecewise constant maps
MSC 2020: 49J45; 49Q05; 49Q15; 28A75

Dedicated to Giovanni Bellettini for his 60th birthday

Communicated by Irene Fonseca

Funding statement: Riccardo Scala is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilitàa e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM), and joins the project CUP-E53C22001930001. We also acknowledge the partial financial support of the F-CUR (project number 2262 – 2022 – SR – CONRICMIURPC – FCUR2022-002) of the University of Siena, and the financial support of PRIN 2022PJ9EFL “Geometric Measure Theory: Structure of Singular Measures, Regularity Theory and Applications in the Calculus of Variations”. The latter has been funded by the European Union under NextGenerationEU (funding organization: Italian Miur, Missione 4, Componente 2 – CUP:E53D23005860006).

A Appendix

We collect here two useful observations. The first one consists in the following lemma, whose content can be found in [19] (see also references therein):

Lemma A.1.

Let U ⊂ R 2 be a relatively compact set. Then for a.e. δ > 0 the δ-neighborhood U δ of U has Lipschitz boundary.

As a second remark, we see that we can equivalently relax the area functional using W loc 1 , ∞ functions instead of C 1 maps:

Lemma A.2.

Let Ω ⊂ R 2 be a Lipschitz domain and u ∈ L 1 ⁢ ( Ω ; R 2 ) . Then

(A.1) 𝒜 ¯ ⁢ ( u , Ω ) = inf ⁡ { lim inf k → + ∞ ⁡ 𝔸 ⁢ ( v k , Ω ) : v k ∈ W loc 1 , ∞ ⁢ ( Ω ; ℝ 2 ) , v k → u ⁢ in  ⁢ L 1 ⁢ ( Ω ; ℝ 2 ) } .

This follows from the fact that for v ∈ W loc 1 , ∞ ⁢ ( Ω ; ℝ 2 ) it holds 𝒜 ¯ ⁢ ( v , Ω ) = 𝔸 ⁢ ( v , Ω ) (see [1]), which trivially implies the inequality ≤ in the formula above. The opposite inequality is obtained by simply observing that

C 1 ⁢ ( Ω ; ℝ 2 ) ⊆ W loc 1 , ∞ ⁢ ( Ω ; ℝ 2 ) .

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Received: 2023-10-08
Accepted: 2024-06-09
Published Online: 2024-10-02
Published in Print: 2025-04-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Boundary behavior of solutions to fractional p-Laplacian equation
  3. Existence of distributional solutions to some quasilinear degenerate elliptic systems with low integrability of the datum
  4. A weakly coupled system of p-Laplace type in a heat conduction problem
  5. On the interior regularity criteria for the viscoelastic fluid system with damping
  6. On the L 1-relaxed area of graphs of BV piecewise constant maps taking three values
  7. On prescribing the number of singular points in a Cosserat-elastic solid
  8. Stability from rigidity via umbilicity
  9. Free boundary regularity in the fully nonlinear parabolic thin obstacle problem
  10. Calderón–Zygmund theory for strongly coupled linear system of nonlocal equations with Hölder-regular coefficient
  11. Inertial (self-)collisions of viscoelastic solids with Lipschitz boundaries
  12. A discrete crystal model in three dimensions: The line-tension limit for dislocations
  13. Two-dimensional graph model for epitaxial crystal growth with adatoms
  14. Global existence for the Willmore flow with boundary via Simon’s Li–Yau inequality
  15. Linearization in magnetoelasticity
  16. The fractional Hopf differential and a weak formulation of stationarity for the half Dirichlet energy
https://doi.org/10.1515/acv-2023-0108
Keywords for this article
Plateau problem; relaxation; minimal connections; area functional; minimal surfaces; piecewise constant maps

Articles in the same Issue

  1. Frontmatter
  2. Boundary behavior of solutions to fractional p-Laplacian equation
  3. Existence of distributional solutions to some quasilinear degenerate elliptic systems with low integrability of the datum
  4. A weakly coupled system of p-Laplace type in a heat conduction problem
  5. On the interior regularity criteria for the viscoelastic fluid system with damping
  6. On the L 1-relaxed area of graphs of BV piecewise constant maps taking three values
  7. On prescribing the number of singular points in a Cosserat-elastic solid
  8. Stability from rigidity via umbilicity
  9. Free boundary regularity in the fully nonlinear parabolic thin obstacle problem
  10. Calderón–Zygmund theory for strongly coupled linear system of nonlocal equations with Hölder-regular coefficient
  11. Inertial (self-)collisions of viscoelastic solids with Lipschitz boundaries
  12. A discrete crystal model in three dimensions: The line-tension limit for dislocations
  13. Two-dimensional graph model for epitaxial crystal growth with adatoms
  14. Global existence for the Willmore flow with boundary via Simon’s Li–Yau inequality
  15. Linearization in magnetoelasticity
  16. The fractional Hopf differential and a weak formulation of stationarity for the half Dirichlet energy
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