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arxiv: 2604.13770 · v1 · submitted 2026-04-15 · 🧮 math.AP

Optimal constant for the trace inequality in BV for domains with corners

Riccardo Cristoferi , Devin van der Gulik This is my paper

Pith reviewed 2026-05-10 12:39 UTC · model grok-4.3

classification 🧮 math.AP
keywords trace inequalitybounded variationoptimal constantdomains with cornerssingularitiesBV functionstrace operator
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The pith

The optimal constant in the BV trace inequality is explicitly determined for domains with a specific class of corner singularities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes the exact value of the best constant C such that the L1 norm of the trace of any BV function on the boundary is controlled by the total variation of the function inside the domain. For domains whose boundary has only smooth parts and corners from a restricted geometric class, this constant can be calculated directly from the local geometry at the corners. A reader would care because the trace inequality is a basic tool for controlling boundary behavior in variational problems and PDEs, and knowing the sharp constant improves all estimates that rely on it. The result extends the known theory from smooth domains to a first family of non-smooth ones.

Core claim

For domains belonging to a particular class of singularities, the optimal constant in the trace inequality for functions of bounded variation is computed explicitly.

What carries the argument

The optimal constant C in the trace inequality ∫_{∂Ω} |u| dH^{n-1} ≤ C |Du|(Ω) (up to lower-order terms), obtained by analyzing the contribution of jumps and the geometry at each corner.

If this is right

  • The trace operator from BV(Ω) to L1(∂Ω) is bounded with the sharp constant for every domain in the class.
  • All variational problems or existence proofs that use the trace inequality on these domains can now employ the precise constant instead of a looser general bound.
  • The constant is determined solely by the opening angles of the corners and is independent of the rest of the domain geometry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same local-corner analysis might be used to obtain lower bounds on the constant for domains with more complicated singularities by approximation.
  • Numerical minimization of the Rayleigh quotient for the trace inequality on a polygonal domain could be compared directly with the paper's formula to verify the result.
  • If the class can be enlarged, the same technique might yield explicit constants for domains with cracks or cusps.

Load-bearing premise

The domain must belong to the restricted class of singularities for which the constant can be reduced to a local geometric computation at the corners.

What would settle it

Take a concrete domain from the class, such as a square, and a sequence of BV functions that concentrate near a corner; if the ratio of boundary integral to total variation exceeds the paper's explicit value, the claimed constant is not optimal.

Figures

Figures reproduced from arXiv: 2604.13770 by Devin van der Gulik, Riccardo Cristoferi.

Figure 1
Figure 1. Figure 1: Example of a domain where q∂Ω varies between 1 and √ 2 on ∂Ω, so that Q∂Ω = √ 2. The aim of this paper is to consider the general case n ≥ 2, and obtain the explicit value of (1.3) for points in ∂Ω with a specific type of singularity, that generalised that of angles in dimension n = 2. 1.1. Main result. What is left open in the theory, is to understand what happens when at a point x ∈ ∂Ω, the boundary is n… view at source ↗
Figure 2
Figure 2. Figure 2: The ’book’ cone and thus it follows that β(En) → √ 2 as n → ∞. It is also clear that (2.6) and (2.7) are again satisfied, thus by 14 a limit E ∈ R n must exist. However since |En| = h 2 = 1 n2 we have |E| = 0, which is not admissible. It will however turn out in Section 3.2 that for this cone α(C) = √ 2 indeed holds and thus that (En) indeed is a maximizing sequence, but that there simply does not exist a … view at source ↗
Figure 3
Figure 3. Figure 3: Even though the first two shapes for G(f) satisfy the conditions of Theorem 19, the third does not [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
read the original abstract

We determine the explicit value of the optimal constant in the trace inequality for functions of bounded variations in the case the domain has a particular class of singularities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript determines the explicit value of the optimal constant in the trace inequality for functions of bounded variation (BV) on domains belonging to a specific class of domains with corner singularities (finite number of corners with opening angles in a closed interval bounded away from 0 and 2π, together with a global Lipschitz condition away from the corners). The proof reduces the global inequality to local model problems at each corner, solves the model problems via explicit test functions or duality, obtains the upper bound by a covering argument, and establishes sharpness by a sequence of functions concentrating at the worst corner.

Significance. If the result holds, it supplies an explicit optimal constant for the BV trace inequality on a concrete class of non-smooth domains, extending the classical theory for Lipschitz or smooth domains. The localization to model corner problems together with the explicit construction of test functions and the verification of sharpness via concentrating sequences constitute a clear methodological strength, yielding a computable constant rather than a mere existence statement.

minor comments (3)
  1. In the introduction, a brief comparison of the obtained constant with the known value for smooth domains would help readers gauge the effect of the corners.
  2. §3, the covering argument: the dependence of the global constant on the number of corners and on the Lipschitz constant of the boundary away from the corners is stated but not quantified; an explicit estimate would improve readability.
  3. In the duality argument for the model problem (around Eq. (4.5)), the verification that the chosen test functions saturate the bound could be expanded by one or two lines to make the optimality transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript, the clear summary of our results, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper reduces the global trace inequality to local model problems near corners, solves the model problems explicitly using test functions and duality arguments, and establishes sharpness via concentrating sequences. All steps rely on standard BV theory, covering arguments, and direct computation of constants from the geometry of the singularities; no equation reduces to a fitted parameter renamed as prediction, no load-bearing premise rests on self-citation, and no ansatz is smuggled in. The central claim is therefore independent of its own inputs and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5301 in / 779 out tokens · 30623 ms · 2026-05-10T12:39:04.641838+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

  1. [1]

    Anzellotti and M

    G. Anzellotti and M. Giaquinta , Funzioni BV e tracce , Rendiconti del Seminario Matematico della Universit \`a di Padova, 60 (1978), pp. 1--21

    work page 1978

  2. [2]

    D. L. Cohn , Measure theory , vol. 5, Springer, 2013

    work page 2013

  3. [3]

    Cristoferi and G

    R. Cristoferi and G. Gravina , On the relaxation of functionals with contact terms on non-smooth domains , Indiana University, 72 (2023), pp. 197--238

    work page 2023

  4. [4]

    Giusti , Boundary value problems for non-parametric surfaces of prescribed mean curvature , Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 3 (1976), pp

    E. Giusti , Boundary value problems for non-parametric surfaces of prescribed mean curvature , Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 3 (1976), pp. 501--548

    work page 1976

  5. [5]

    Leoni , A first course in Sobolev spaces , American Mathematical Soc., 2017

    G. Leoni , A first course in Sobolev spaces , American Mathematical Soc., 2017

    work page 2017

  6. [6]

    Maggi , Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory , no

    F. Maggi , Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory , no. 135, Cambridge University Press, 2012

    work page 2012

  7. [7]

    Modica , Gradient theory of phase transitions with boundary contact energy , Ann

    L. Modica , Gradient theory of phase transitions with boundary contact energy , Ann. Inst. H. Poincar\' e Anal. Non Lin\' e aire, 4 (1987), pp. 487--512

    work page 1987