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arxiv: 2604.05245 · v2 · submitted 2026-04-06 · 🧮 math.AP

The two-phase Alt-Phillips problem for quasilinear operators

Yousef Alamri , Jos\'e Miguel Urbano This is my paper

Pith reviewed 2026-05-10 18:37 UTC · model grok-4.3

classification 🧮 math.AP
keywords quasilinear operatorsAlt-Phillips functionaltwo-phase free boundariesinterior regularityoptimal growthrectifiabilitydensity estimatesp-Laplacian type operators
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The pith

Sign-changing minimizers of the quasilinear Alt-Phillips functional have interior regularity and optimal growth for all 1<p<∞ and 0<γ<p.

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

This paper establishes interior regularity together with optimal growth estimates near the zero set for sign-changing minimizers of a quasilinear version of the Alt-Phillips functional, covering the complete range of the exponent p and the nonlinearity power γ. It further derives local finite-perimeter and density estimates on the free boundary, which in turn yield local rectifiability and finite Hausdorff measure of the reduced and two-phase free boundaries when γ belongs to a suitable sub-range. A reader would care because the results apply to a broad family of nonlinear elliptic equations that arise in two-phase models, supplying uniform control on how the positive and negative phases meet without extra restrictions on the operator beyond the stated ellipticity and growth.

Core claim

Interior regularity and optimal growth estimates hold for sign-changing minimizers of the p-singular or p-degenerate quasilinear Alt-Phillips functional throughout the full range 1<p<∞ and 0<γ<p. Local finite-perimeter and density estimates are obtained, from which the local (N-1)-rectifiability of the reduced and two-phase free boundaries and the local finiteness of their (N-1)-dimensional Hausdorff measure follow for a restricted range of γ.

What carries the argument

The quasilinear Alt-Phillips functional whose Euler-Lagrange equation is driven by a p-singular or p-degenerate operator, with sign-changing minimizers whose nodal set forms the two-phase free boundary.

If this is right

  • Sign-changing minimizers are regular in the interior of the domain.
  • Optimal growth estimates hold near the nodal set.
  • The free boundary has locally finite perimeter and satisfies density estimates.
  • For a restricted range of γ the reduced and two-phase free boundaries are locally (N-1)-rectifiable with locally finite (N-1)-Hausdorff measure.

Where Pith is reading between the lines

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

  • The density estimates could be combined with monotonicity formulas to study the topology of the free boundary in higher dimensions.
  • The same growth-control techniques might adapt to quasilinear operators with variable coefficients or to obstacle-type problems.
  • Rectifiability opens the possibility of further regularity results once additional smoothness assumptions are placed on the operator.

Load-bearing premise

The quasilinear operator satisfies uniform ellipticity and growth conditions that permit the existence of sign-changing minimizers whose behavior is controlled by the stated functional.

What would settle it

An explicit sign-changing minimizer for which the growth near the zero set fails to be optimal, or whose reduced free boundary fails to be (N-1)-rectifiable inside the allowed range of γ, would disprove the claims.

read the original abstract

We establish interior regularity and optimal growth estimates for sign-changing minimizers of the $p-$singular or $p-$degenerate quasilinear Alt--Phillips functional throughout the full range of $1<p<\infty$ and of the nonlinearity power $0<\gamma<p$. In addition, we obtain local finite perimeter and density estimates, from which we deduce the local $(N-1)$-rectifiability of the reduced and two-phase free boundaries and the local finiteness of their $(N-1)$-dimensional Hausdorff measure for a restricted range of $\gamma$.

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 establishes interior regularity and optimal growth estimates for sign-changing minimizers of the p-singular or p-degenerate quasilinear Alt-Phillips functional throughout the full range 1 < p < ∞ and 0 < γ < p. It additionally obtains local finite perimeter and density estimates, from which it deduces the local (N-1)-rectifiability of the reduced and two-phase free boundaries together with the local finiteness of their (N-1)-dimensional Hausdorff measure, but only for a restricted range of γ.

Significance. If the results hold, this work extends the classical Alt-Phillips theory to a broad class of quasilinear operators, covering both singular and degenerate cases without extra restrictions on the parameters. The direct analytic proof via variational methods and standard elliptic regularity theory is a clear strength, as is the explicit separation between the full-range analytic properties and the restricted-range geometric conclusions. The manuscript thereby supplies a parameter-free derivation of the stated estimates under the given ellipticity and growth conditions on the operator.

minor comments (3)
  1. The abstract states that the rectifiability and Hausdorff measure results hold 'for a restricted range of γ' but does not indicate the precise interval; adding this detail would improve immediate readability.
  2. Notation for the quasilinear operator (including the precise ellipticity and growth assumptions) is introduced somewhat late; moving a concise statement of these structural hypotheses to the introduction or §2 would aid readers.
  3. A few typographical inconsistencies appear in the indexing of free-boundary sets (e.g., the use of 'reduced' versus 'two-phase' free boundary in §4 and §5); uniformizing the terminology would enhance clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for noting the clear separation between the full-range interior regularity and growth estimates and the restricted-range geometric conclusions on free-boundary rectifiability. We appreciate the recommendation for minor revision and are pleased that the direct variational approach and parameter-free nature of the results were viewed as strengths.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes interior regularity, optimal growth estimates, and rectifiability results for minimizers of the quasilinear Alt-Phillips functional via variational methods and standard elliptic regularity theory over the full parameter ranges 1 < p < ∞ and 0 < γ < p. The derivation chain relies on known PDE techniques without self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse claims to inputs by construction. Central results on free boundary properties are obtained from density estimates and perimeter theory in a restricted γ subrange, with explicit separation of arguments and no internal equivalence to the stated assumptions on the operator or minimizers.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract, the paper rests on standard assumptions from variational PDE theory; no free parameters, new entities, or ad-hoc axioms are introduced.

axioms (2)
  • domain assumption The quasilinear operator satisfies standard ellipticity and growth conditions
    Required to define the functional and apply regularity theory.
  • domain assumption Minimizers of the functional exist and are sign-changing
    The statements concern properties of such minimizers.

pith-pipeline@v0.9.0 · 5381 in / 1322 out tokens · 46910 ms · 2026-05-10T18:37:41.353408+00:00 · methodology

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Reference graph

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6 extracted references · 6 canonical work pages

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