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arxiv: 2605.23835 · v1 · pith:VHRJJRDOnew · submitted 2026-05-22 · 🧮 math.AP · math.FA

Pointwise Estimates Near Singular Sets for Quasilinear Elliptic Equations

Juan Pablo Alcon Apaza This is my paper

Pith reviewed 2026-05-25 03:16 UTC · model grok-4.3

classification 🧮 math.AP math.FA
keywords quasilinear elliptic equationsvariable exponentssingular setsremovabilityFinsler manifoldspointwise estimates1-Laplacian
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The pith

A pointwise estimate |u(x)| ≤ C ρ1(x)^{-τ} near Γ proves removability of singularities for quasilinear elliptic equations and yields convergence to 1-Laplacian solutions.

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

The paper derives the bound |u(x)| ≤ C ρ1(x)^{-τ} near the singular set Γ for weak solutions of -Δ_{p(x)} u + |u|^{q-1} u =0 in domains of a Finsler manifold. This bound serves as the key tool to establish that the singularity at Γ is removable. Under the existence assumption for solutions at each variable exponent p with 1 < p- ≤ p+ < min{2, q+1}, a subsequence converges as p+ → 1 to a BV solution of the limiting equation -Δ_1 u + |u|^{q-1} u =0. A sympathetic reader would care because the result controls solution behavior near singularities while passing from variable-exponent p-Laplace problems to the degenerate 1-Laplace case.

Core claim

The central claim is that the estimate |u(x)| ≤ C ρ1(x)^{-τ} holds near Γ for weak solutions u in W^{1,p(x)}_{loc}(Ω̄∖(Γ∪Σ);ϑ) ∩ L^∞_{loc} away from Γ and Σ, with C and τ approaching positive values as p+ →1; this estimate is the key ingredient proving the singularity at Γ is removable. Moreover, assuming existence of u_p for 1<p-≤p+<min{2,q+1}, there is subsequence convergence as p_m+→1 to a solution u∈BV(U;ϑ)∩L^{q+1}(U;ϑ) of -Δ_1 u + |u|^{q-1}u=0 in U.

What carries the argument

The pointwise estimate |u(x)| ≤ C ρ1(x)^{-τ} near Γ, derived from Lipschitz distance-type functions ρ1 and ρ2 satisfying F(·,∇ρ1)≤1 and F(·,∇ρ2)≤1 a.e.

If this is right

  • The singularity at Γ is removable for the weak solutions of the quasilinear equation.
  • There exists a subsequence {u_{p_m}} with p_m+→1 converging to a solution u in BV(U;ϑ)∩L^{q+1}(U;ϑ) of the 1-Laplacian equation.
  • The constants C and τ in the pointwise estimate remain positive in the limit p+→1.
  • The removability result holds in bounded domains Ω of the Finsler manifold (M,F,ϑ).
  • The estimate applies to the model problem in R^n with ρ1(x)=|(x_{d+1},…,x_n)| and ρ2(x)=|(x_1,…,x_d)|.

Where Pith is reading between the lines

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

  • The convergence supplies a limiting procedure that approximates 1-Laplace problems by variable-exponent p-Laplace problems even when singular sets are present.
  • The technique of using distance-type Lipschitz functions ρ1, ρ2 may connect to regularity questions for other quasilinear equations with isolated singular sets.
  • The Finsler-manifold setting indicates the removability conclusion is stable under changes from Euclidean to non-Euclidean geometry.

Load-bearing premise

The assumption that for every variable exponent satisfying 1<p-≤p+<min{2,q+1} there exists a weak solution u_p in W^{1,p(x)}_{loc}(Ω;ϑ)∩L^∞_{loc}(Ω) of the equation in the bounded domain Ω.

What would settle it

A weak solution u near Γ that violates |u(x)| ≤ C ρ1(x)^{-τ} for every C>0 and τ>0, or a sequence u_p with p+→1 that fails to converge to any BV solution of -Δ_1 u + |u|^{q-1}u=0.

read the original abstract

In this work, we study the removability of boundary singular sets for certain classes of quasilinear elliptic equations in domains $\Omega$ of an $n$-dimensional Finsler manifold ( $\mathcal{M}, F, \vartheta$ ). We work with Lipschitz functions $\rho_1$ and $\rho_2$ satisfying distance-type properties; in particular, $F(\cdot, \boldsymbol{\nabla} \rho_1) \leq 1$ and $F(\cdot, \boldsymbol{\nabla} \rho_2) \leq 1$ a.e. in $\mathcal{M}$. The singular set is defined by $\Gamma=\rho_1^{-1}(\{0\})$. The model problem is $-\Delta_{p(x)} u+|u|^{q-1} u=0$ in domains of $\mathbb{R}^n \cong \mathbb{R}^d \times \mathbb{R}^{n-d} \cong \rho_1^{-1}(\{0\}) \times \rho_2^{-1}(\{0\})$, where $\rho_1(x)=|(x_{d+1}, \ldots, x_n)|$ and $\rho_2(x)=|(x_1, \ldots, x_d)|$. The main tool in our analysis is the estimate $$ |u(x)| \leq \mathbf{C} \rho_1(x)^{-\tau} $$ near $\Gamma$ for weak solutions $u \in W_{loc}^{1, p(x)}(\bar{\Omega} \backslash(\Gamma \cup \Sigma) ; \vartheta) \cap L_{loc}^{\infty}(\bar{\Omega} \backslash(\Gamma \cup \Sigma))$, where the constants $\mathbf{C}>0$ and $\tau>0$ converge to positive values as $p^{+} \rightarrow 1$. This estimate is a key ingredient in proving that the singularity at $\Gamma$ is removable. Moreover, in a bounded domain $\Omega$, using this estimate and assuming that, for every variable exponent satisfying $1<p^{-} \leq p^{+}<\min \{2, q+1\}$, there exists a weak solution $u_p \in W_{loc}^{1, p(x)}(\Omega ; \vartheta) \cap L_{loc}^{\infty}(\Omega)$ of $$ -\operatorname{div}\left(|\boldsymbol{\nabla} u_p|_F^{p-2} \boldsymbol{\nabla} u_p\right)+|u_p|^{q-1} u_p=0 \quad \text { in } \Omega, $$ we prove that, for every $U \Subset \Omega$, there exists a subsequence $\{u_{p_m}\}$, with $p_m^{+} \rightarrow 1$, that converges to a solution $u \in B V(U ; \vartheta) \cap L^{q+1}(U ; \vartheta)$ of $$ -\Delta_1 u+|u|^{q-1} u=0 \quad \text { in } U . $$

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

1 major / 1 minor

Summary. The manuscript claims to establish the pointwise estimate |u(x)| ≤ C ρ_1(x)^{-τ} near the singular set Γ=ρ_1^{-1}({0}) for weak solutions u ∈ W^{1,p(x)}_{loc}(Ω̄∖(Γ∪Σ);ϑ) ∩ L^∞_{loc} of -Δ_{p(x)}u + |u|^{q-1}u=0 on Finsler manifolds, with C,τ>0 as p^+→1; this estimate is a key ingredient for removability of singularities at Γ. Additionally, assuming existence of solutions u_p ∈ W^{1,p(x)}_{loc}(Ω;ϑ)∩L^∞_{loc}(Ω) for every 1<p^-≤p^+<min{2,q+1}, it claims subsequence convergence as p_m^+→1 to a solution u∈BV(U;ϑ)∩L^{q+1}(U;ϑ) of -Δ_1 u + |u|^{q-1}u=0 in bounded subdomains U.

Significance. If valid, the pointwise estimate supplies a concrete decay rate near codimension-(n-d) singular sets that could support removability arguments for variable-exponent quasilinear equations on Finsler manifolds. The conditional convergence result would connect the variable-exponent theory to the 1-Laplacian limit, but its value is limited by the explicit existence hypothesis on which it rests. No machine-checked proofs or parameter-free derivations are present.

major comments (1)
  1. [Abstract] Abstract (and the corresponding convergence section): the subsequence convergence claim as p_m^+→1 is load-bearing on the explicit hypothesis that weak solutions u_p exist for every variable exponent satisfying 1<p^-≤p^+<min{2,q+1}. The manuscript states this existence as an assumption rather than deriving or citing it, so the convergence statement is void without independent verification of the hypothesis.
minor comments (1)
  1. [Abstract] The notation for the constants C and τ (boldface C in the abstract) and the precise dependence of τ on p^+, q, n, d should be clarified in the statement of the main estimate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review. The sole major comment is addressed point-by-point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the corresponding convergence section): the subsequence convergence claim as p_m^+→1 is load-bearing on the explicit hypothesis that weak solutions u_p exist for every variable exponent satisfying 1<p^-≤p^+<min{2,q+1}. The manuscript states this existence as an assumption rather than deriving or citing it, so the convergence statement is void without independent verification of the hypothesis.

    Authors: We agree that the convergence result is conditional on the stated existence hypothesis for u_p. This hypothesis is explicitly declared in both the abstract and the convergence section, and the theorem is formulated accordingly as a conditional statement. The manuscript's primary contributions are the pointwise decay estimate |u| ≤ C ρ_1^{-τ} (with constants controlled as p^+ → 1) and its application to removability; the convergence to the 1-Laplacian is presented as a consequence once existence is granted. Establishing existence of weak solutions to the variable-exponent equation on Finsler manifolds for the full range 1 < p^- ≤ p^+ < min{2,q+1} is a substantial analytic task outside the scope of the present work, which centers on a priori estimates near singular sets. No citation for this existence result is known to us in the precise setting considered, and we therefore retain the assumption as written. The conditional nature of the claim is already transparent to readers. revision: no

Circularity Check

0 steps flagged

No circularity; estimate derived from weak form and convergence conditional on explicit external assumption

full rationale

The paper presents the pointwise estimate as obtained from analysis of the weak formulation of -Δ_{p(x)} u + |u|^{q-1}u = 0 for any given weak solution in the stated Sobolev class. The subsequence convergence to a 1-Laplacian solution is stated only under the separate hypothesis that solutions u_p exist for every admissible variable exponent; the paper does not derive or claim to derive that existence. No equations reduce by construction to fitted inputs, no self-citations are load-bearing for the central claims, and no ansatz or uniqueness result is smuggled in. The derivation chain is therefore self-contained against the paper's own stated assumptions and does not collapse to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on abstract; relies on standard background in variable-exponent Sobolev spaces, Finsler geometry, and weak solutions, plus one explicit existence assumption for the convergence step. No free parameters or invented entities are visible.

axioms (2)
  • domain assumption Finsler manifold (M,F,ϑ) and Lipschitz ρ1,ρ2 satisfy F(·,∇ρi)≤1 a.e. with distance-type properties
    Used to define Γ=ρ1^{-1}(0) and set up the model problem in R^n ≅ R^d × R^{n-d}.
  • domain assumption Existence of weak solutions u_p for every p(x) with 1<p−≤p+<min{2,q+1}
    Explicitly assumed to obtain the p+→1 convergence to the 1-Laplacian solution in BV.

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