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arxiv: 2605.03322 · v1 · submitted 2026-05-05 · 🧮 math.AP · math.PR

Explosion versus decay for boundary derivatives of p-harmonic functions as p tends to 1: nonlocality

Yuval Peres , Han Wang This is my paper

Pith reviewed 2026-05-07 15:41 UTC · model grok-4.3

classification 🧮 math.AP math.PR
keywords p-Laplacianboundary derivativesp-harmonic functionsnonlocalityDirichlet problemasymptotics as p to 1Lipschitz domains
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The pith

The boundary derivative of p-harmonic functions with 0-1 data either explodes at rate C/(p-1) or decays exponentially as p approaches 1, depending on nonlocal features of the domain.

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

The paper examines the Dirichlet problem for the p-Laplacian on bounded Lipschitz domains with boundary data taking only the values 0 and 1. It supplies sufficient conditions under which the normal derivative at a boundary point explodes proportionally to 1/(p-1) and other conditions under which it decays exponentially fast with rate exp(-c/(p-1)). The decisive factor is not local geometry near the point but global properties of the whole domain. A cylindrical domain is treated separately and yields the intermediate explosion rate 1/sqrt(p-1).

Core claim

For the Dirichlet problem for the p-Laplacian with {0,1}-valued boundary data on a bounded Lipschitz domain Ω, the boundary derivative explodes at rate C_Ω/(p-1) under certain conditions and decays at rate exp(−c_Ω/(p-1)) under others as p ↓ 1; whether explosion or decay occurs is determined nonlocally by Ω. In a critical cylindrical example the derivative explodes at rate C_d/√(p-1).

What carries the argument

The normal derivative at the boundary of the p-harmonic function, whose rate of growth or decay as p ↓ 1 is controlled by global domain geometry rather than local boundary data.

If this is right

  • Local boundary regularity alone cannot determine the asymptotic rate of the derivative.
  • Global domain features must be taken into account when predicting limiting behavior as p tends to 1.
  • In cylindrical geometries the explosion rate slows to order 1/sqrt(p-1).
  • The nonlocality persists for all bounded Lipschitz domains equipped with exactly binary boundary data.

Where Pith is reading between the lines

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

  • Similar nonlocal dependence on domain geometry may appear in other degenerate elliptic equations or in the 1-Laplacian limit.
  • Numerical schemes for small p-1 may need to incorporate global information rather than local stencils alone.
  • The dichotomy could guide the design of test domains for studying the transition from p-Laplacian to total-variation problems.

Load-bearing premise

The domain is bounded and Lipschitz and the boundary data takes exactly the values 0 and 1.

What would settle it

A bounded Lipschitz domain with 0-1 boundary data in which the observed boundary derivative rate as p ↓ 1 is neither proportional to 1/(p-1), nor exponentially decaying, nor proportional to 1/sqrt(p-1) would falsify the stated dichotomy and rates.

Figures

Figures reproduced from arXiv: 2605.03322 by Han Wang, Yuval Peres.

Figure 1
Figure 1. Figure 1: The behavior of radial p-harmonic function for different p. As p approaches 1, the boundary derivative grows rapidly near {∣x∣ = 1}, while it approaches 0 near {∣x∣ = 2}. 1 arXiv:2605.03322v1 [math.AP] 5 May 2026 view at source ↗
Figure 2
Figure 2. Figure 2: The (absolute value) of the boundary derivative at the boundary as a function of p. The natural question is how to classify such behaviors. From the example, a naive guess is that the curvature of the boundary determines them. However, this is not the complete story. As we shall see later, the limiting behavior of the normal derivative is actually determined by some nonlocal conditions on the boundary valu… view at source ↗
Figure 3
Figure 3. Figure 3: If {F = 0}∖{x0} is contained in a ball (with red boundary) tangent at x0, then a lower bound of order 1 p−1 on the boundary derivative holds as p ↓ 1; see Theorem 1.1. On the other hand, the condition of exponential decay in Theorem 1.2 is about {F = 1} being separated by a hyperplane from x0. See figure 4. The exact definitions and proof of the results are in Section 2 and 3. Ω x0 F = 0 F = 1 view at source ↗
Figure 4
Figure 4. Figure 4: If {F = 1} is separated by a hyperplane from x0, then the exponential decay of boundary derivative holds in Theorem 1.2. The curvature at x0 is irrelevant. A special method using measure transformations also appears in the probabilistic approach. The intermediate result in this approach, Lemma 3.1, is also of its own interest, as it gives a lower bound (with explicit dependence on p and d) for the hitting … view at source ↗
Figure 5
Figure 5. Figure 5: The cylinder and the boundary condition in Theorem 1.3. F = 1 on the top and sides of the cylinder. Theorem 1.3. Consider the cylinder Q = {(x, y) ∶ x ∈ B d (0, 1), y ∈ (0, 1)} ⊂ R d+1 depicted in view at source ↗
Figure 6
Figure 6. Figure 6: The test function v for the lower bound in (b). Upper bound in (a): We then show an upper bound for any general domain. The strategy is similar. Since the boundary is locally C 2 , it satisfies the exterior ball condition; hence we can find a ball B(−δe1, δ) such that B(−δe1, δ) ∩ Ω = {0}. We then pick δ ′ small enough so that ∂Ω ∩ {F = 1} ∩ B(−δe1, δ + δ ′ ) = ∅. Again we let w be the unique p-harmonic fu… view at source ↗
Figure 7
Figure 7. Figure 7: The test function w for the upper bound in (a) view at source ↗
Figure 8
Figure 8. Figure 8: The boundary condition in Lemma 3.1. By the comparison principle, we may prove the estimate for another p-harmonic function v with a continuous boundary condition F ∶ ∂Q → R, where F(x, y) = ⎧⎪⎪ ⎨ ⎪⎪⎩ 0, y = 0 or ∣x∣ = 1, min{1, 10 − 10∣x∣}, y = H and ∣x∣ < 1. (3.3) Now we consider the tug-of-war with noise on Q = B d−1 (0, 1)×(0, H), with initial state (0, h), boundary condition F, and sufficiently small … view at source ↗
Figure 9
Figure 9. Figure 9: In particular, Qi ⊂ Ω by our definition of δ. xi xi+1 xi+2 Qi+1 Qi view at source ↗
read the original abstract

We consider the Dirichlet problem for the $p$-Laplacian on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ with a $\{0,1\}$-valued function as the boundary condition and study the dependence of the boundary derivative on $p$ as $p\downarrow1$. We provide sufficient conditions for the derivative to explode at rate $\frac{C_\Omega}{p-1}$ and to decay at rate $\exp(-\frac{c_\Omega}{p-1})$. Surprisingly, whether explosion or decay occurs is not determined locally. We also present a critical example of a cylinder where this derivative explodes at rate $\frac{C_d}{\sqrt{p-1}}$.

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

2 major / 2 minor

Summary. The manuscript studies the Dirichlet problem for the p-Laplacian on a bounded Lipschitz domain Ω ⊂ R^d with {0,1}-valued boundary data. It derives sufficient conditions under which the boundary normal derivative of the p-harmonic function either explodes at rate C_Ω/(p−1) or decays at rate exp(−c_Ω/(p−1)) as p ↓ 1. The choice between these regimes is shown to depend on global properties of Ω rather than local boundary geometry. A critical example on a cylinder is given where the derivative explodes at the intermediate rate C_d/√(p−1).

Significance. If the claims hold, the work provides quantitative insight into the nonlocal character of the p → 1 limit, connecting p-harmonic functions to the 1-Laplacian and total-variation minimization. The explicit rates and the demonstration that local geometry is insufficient to determine the asymptotics are potentially useful for both theoretical analysis of BV problems and numerical schemes that approximate the 1-Laplacian. The cylinder example supplies a concrete benchmark case with a distinct scaling.

major comments (2)
  1. [Main results / Theorems] The sufficient conditions for explosion versus decay (presumably stated in the main theorems) are only sketched in the abstract; the precise geometric or analytic hypotheses on Ω that distinguish the two regimes must be stated explicitly, including any dependence on the Lipschitz constant of ∂Ω, so that the nonlocality claim can be verified by comparing domains that agree locally near a boundary point but differ globally.
  2. [Cylinder example] In the cylinder example, the claimed rate C_d/√(p−1) differs from the generic C_Ω/(p−1) rate; the derivation must clarify whether this intermediate scaling arises from the infinite extent in one direction or from a specific interaction between the boundary data and the p-Laplacian degeneracy, and whether the constant C_d is independent of the cylinder radius.
minor comments (2)
  1. [Abstract / Introduction] Notation for the boundary derivative (normal derivative of the p-harmonic function) should be introduced once and used consistently; currently the abstract refers to “the derivative” without specifying the point or the direction.
  2. [Main results] The dependence of the constants C_Ω and c_Ω on the domain should be tracked more explicitly in the statements, even if only qualitatively, to facilitate comparison with the cylinder case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. We address each major comment below and will make the necessary revisions to improve the clarity of the presentation.

read point-by-point responses

  1. Referee: The sufficient conditions for explosion versus decay (presumably stated in the main theorems) are only sketched in the abstract; the precise geometric or analytic hypotheses on Ω that distinguish the two regimes must be stated explicitly, including any dependence on the Lipschitz constant of ∂Ω, so that the nonlocality claim can be verified by comparing domains that agree locally near a boundary point but differ globally.

    Authors: We agree that the abstract only sketches the conditions. In the revised version, we will explicitly formulate the sufficient conditions in the statements of the main theorems, including the precise hypotheses on the domain Ω. These conditions are based on the existence of suitable barrier functions or the positivity of certain nonlocal quantities derived from the geometry of Ω. The constants C_Ω and c_Ω depend on the Lipschitz constant of the boundary, but the choice of regime (explosion or decay) is governed by global features, such as the presence of paths or sets connecting different parts of the boundary. To illustrate the nonlocality, we will include a brief discussion or example of two Lipschitz domains that coincide in a neighborhood of the boundary point of interest but differ elsewhere, leading to different asymptotic behaviors. This addresses the verification request. revision: yes

  2. Referee: In the cylinder example, the claimed rate C_d/√(p−1) differs from the generic C_Ω/(p−1) rate; the derivation must clarify whether this intermediate scaling arises from the infinite extent in one direction or from a specific interaction between the boundary data and the p-Laplacian degeneracy, and whether the constant C_d is independent of the cylinder radius.

    Authors: The cylinder example in Section 4 is a bounded domain of the form Ω = (0,1) × B(0,R) ⊂ R^d, with boundary data equal to 1 on one end and 0 on the lateral surface and the other end. The intermediate rate C_d / √(p-1) arises from the specific interaction between the flat boundary portions and the degeneracy of the p-Laplacian as p approaches 1, rather than any infinite extent (the domain is bounded). The derivation involves a careful scaling analysis and comparison with one-dimensional p-harmonic functions in the axial direction. The constant C_d depends only on the dimension d and is independent of the radius R, as the scaling is chosen such that the radial contribution becomes negligible in the limit. We will add a clarifying paragraph in the revised manuscript to explicitly state these points and remove any potential ambiguity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via PDE analysis

full rationale

The paper derives sufficient conditions for boundary derivative explosion at rate C_Ω/(p-1) or decay at exp(-c_Ω/(p-1)) as p↓1, along with nonlocality and the cylinder critical example at C_d/√(p-1), directly from analysis of the p-Laplacian Dirichlet problem on bounded Lipschitz domains with {0,1}-valued boundary data. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the rates and nonlocality follow from the global character of the operator and limiting 1-Laplacian behavior without renaming known results or smuggling ansatzes. The claims remain internally consistent with the stated assumptions and do not rely on prior author work as an unverified uniqueness theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Dirichlet problem for the p-Laplacian being well-posed on bounded Lipschitz domains with binary boundary data; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • domain assumption Ω is a bounded Lipschitz domain in R^d
    Required for the Dirichlet problem setup stated in the abstract.
  • domain assumption Boundary condition takes values in {0,1}
    Used to study the p-dependence of the boundary derivative.

pith-pipeline@v0.9.0 · 5415 in / 1233 out tokens · 70994 ms · 2026-05-07T15:41:33.518671+00:00 · methodology

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Reference graph

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