Essential spectrum for the $p-$Laplacian

Giovanni Franzina; Lorenzo Brasco; Luca Briani

arxiv: 2605.20488 · v1 · pith:AVDCWIVMnew · submitted 2026-05-19 · 🧮 math.AP · math.SP

Essential spectrum for the p-Laplacian

Lorenzo Brasco , Luca Briani , Giovanni Franzina This is my paper

Pith reviewed 2026-05-21 06:35 UTC · model grok-4.3

classification 🧮 math.AP math.SP

keywords essential spectrump-LaplacianPersson theoremPoincaré constantunbounded domainsvariational methodsDirichlet problem

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The pith

The bottom of the essential spectrum for the Dirichlet p-Laplacian equals the sharp Poincaré constant at infinity.

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

This paper defines a variational notion of the essential spectrum for the p-Laplacian on unbounded domains. It extends Persson's classical theorem from the linear case to this nonlinear setting. The result gives a geometric way to find the lowest point in the essential spectrum by measuring the best constant for the L^p Poincaré inequality far from the origin. When p equals 2 the new definition agrees exactly with the usual linear essential spectrum. The authors also give an example on a strip domain where the entire spectrum is essential with no discrete eigenvalues mixed in.

Core claim

We introduce a variational notion of essential spectrum for the Dirichlet p-Laplacian. This allows us to extend Persson's theorem, showing that the bottom of the essential spectrum is equal to the limit as R goes to infinity of the infimum of the Rayleigh quotient over test functions vanishing inside the ball of radius R. This limit is precisely the sharp L^p Poincaré constant at infinity. For p=2 the construction coincides with the classical one, and on a rectilinear strip the spectrum is purely essential.

What carries the argument

The variational essential spectrum defined through the Rayleigh quotient on functions with support escaping to infinity, which extends the classical Persson characterization to the p-Laplacian.

If this is right

The bottom of the essential spectrum depends only on the geometry of the domain at large distances.
For any p greater than 1 the essential spectrum starts at the same value as the infimum of the spectrum on the domain truncated far away.
The spectrum on a rectilinear strip consists only of essential spectrum with no embedded eigenvalues.
The proofs rely on elementary variational arguments that work uniformly for all p.

Where Pith is reading between the lines

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

This variational approach could be applied to other nonlinear eigenvalue problems to locate their essential spectra.
Local perturbations of the domain far from infinity should leave the essential spectrum unchanged.
The constant at infinity might help detect gaps or the absence of discrete eigenvalues in more complex unbounded domains.

Load-bearing premise

The variational formulation on unbounded domains admits a well-defined notion of behavior at infinity that correctly captures the essential spectrum.

What would settle it

A domain where the limit of the infimum of the Rayleigh quotient outside large balls differs from the bottom of the essential spectrum computed via other means, such as Weyl sequences when p equals 2.

read the original abstract

We introduce a variational notion of essential spectrum for the Dirichlet $p-$Laplacian. We then extend the classical Persson Theorem to this nonlinear setting. This result provides a geometric characterization of the bottom of the essential spectrum, in terms of the sharp $L^p$ Poincar\'e constant ``at infinity''. We also show that in the case $p=2$ our construction of the essential spectrum is perfectly consistent with the classical theory. Finally, as an example, we compute the full spectrum of the Dirichlet $p-$Laplacian on a rectilinear strip: it is purely essential, with no embedded eigenvalues. The arguments of the proofs are elementary and new already for the linear case $p=2$.

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.

Desk Editor's Note private letter to a colleague

The paper defines a variational notion of essential spectrum for the p-Laplacian and extends Persson's theorem to characterize its bottom via the Poincaré constant at infinity, with checks for p=2.

read the letter

The main thing here is a variational definition of the essential spectrum for the Dirichlet p-Laplacian, along with an extension of Persson's theorem that characterizes the bottom of this spectrum using the sharp L^p Poincaré constant at infinity. They also confirm consistency with the linear case when p equals 2. The paper does well by using elementary arguments and providing a concrete example on a rectilinear strip. In that case, the spectrum is purely essential with no embedded eigenvalues, which helps verify that the new notion works as intended and captures the behavior at infinity properly. The soft spots are limited. The key assumption is that the variational formulation correctly identifies the essential spectrum through the Poincaré constant at infinity, and nothing in the material suggests this fails for p not equal to 2. The claims of novelty even for the linear case are interesting, but the details would need checking to ensure no reliance on prior results in a way that undermines the extension. Overall, the evidence from the abstract and stress test supports the central claims without major issues. This paper targets researchers in spectral theory for nonlinear operators on unbounded domains. A reader interested in generalizing classical results like Persson's theorem to p-Laplacians would find it worthwhile. It has enough substance and grounding to deserve a serious referee. I would recommend sending this to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces a variational notion of essential spectrum for the Dirichlet p-Laplacian on unbounded domains. It extends the classical Persson theorem to this nonlinear setting, yielding a geometric characterization of the bottom of the essential spectrum in terms of the sharp L^p Poincaré constant at infinity. The construction is shown to be consistent with the classical linear theory when p=2, and the full spectrum is computed explicitly on a rectilinear strip, where it is purely essential with no embedded eigenvalues. The proofs are presented as elementary.

Significance. If the central results hold, the work supplies a useful variational extension of essential-spectrum theory to the p-Laplacian, with a concrete geometric characterization that may aid analysis of nonlinear problems on non-compact domains. The explicit verification for p=2 and the rectilinear-strip example provide direct support for the framework. The elementary character of the arguments, even in the linear case, is a notable strength.

minor comments (2)

The abstract states that the arguments are 'elementary and new already for the linear case p=2'; a short paragraph in the introduction comparing the new variational approach with classical proofs of Persson's theorem would help readers appreciate the novelty.
In the rectilinear-strip example, the computation of the sharp L^p Poincaré constant at infinity is presented as direct verification; adding an explicit formula or a brief derivation of this constant would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive report, including the recommendation for minor revision. We appreciate the recognition of the variational extension of Persson's theorem to the p-Laplacian and the consistency checks provided in the manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a new variational notion of essential spectrum for the Dirichlet p-Laplacian and extends the classical Persson theorem to characterize its bottom via the sharp L^p Poincaré constant at infinity. The abstract explicitly states consistency with the linear case p=2 and that the arguments are elementary and new even for p=2. No load-bearing derivation step reduces by construction to a fitted input, self-citation chain, or definitional equivalence; the geometric characterization and the rectilinear strip example function as independent verification rather than tautological renaming. The construction is presented as self-contained against external benchmarks from classical linear theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces a new variational definition rather than relying on many additional free parameters or invented entities; it rests primarily on standard functional-analytic assumptions for the p-Laplacian.

axioms (1)

standard math Standard variational formulation and Sobolev-space properties of the Dirichlet p-Laplacian on unbounded domains
The extension of Persson's theorem presupposes the usual energy functional and weak formulation for the p-Laplacian.

pith-pipeline@v0.9.0 · 5643 in / 1253 out tokens · 47156 ms · 2026-05-21T06:35:43.265850+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We define the variational essential spectrum ... as the set of λ admitting singular constrained Palais-Smale sequences at level λ (Definition 1.6). Main Theorem: Ep(Ω) = inf Sess,p(Ω).
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Proof constructs sequence via Vn repulsive + εn confining potentials and obtains almost-critical points at Ep(Ω).

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

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G. Teschl,Mathematical methods in quantum mechanics. With applications to Schr¨ odinger operators.Second edition. Graduate Studies in Mathematics,157. American Mathematical Society, Providence, RI, 2014. 2, 6

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K. J. Witsch, Examples of embedded eigenvalues for the Dirichlet-Laplacian in domains with infinite boundaries, Math. Methods Appl. Sci.,12(1990), 177–182. 5, 7 (L. Brasco)Dipartimento di Matematica e Informatica Universit`a degli Studi di Ferrara Via Machiavelli 35, 44121 Ferrara, Italy Email address:lorenzo.brasco@unife.it (L. Briani)School of Computati...

work page 1990

[1] [1]

Anane, Simplicit´ e et isolation de la premi` ere valeur propre dup−laplacien avec poids, C

A. Anane, Simplicit´ e et isolation de la premi` ere valeur propre dup−laplacien avec poids, C. R. Acad. Sci. Paris S´ er. I Math.,305(1987), 725–728. 3

work page 1987

[2] [2]

C. J. Amick, J. F. Toland, Nonlinear elliptic eigenvalue problems on an infinite strip – global theory of bifur- cation and asymptotic bifurcation, Math. Ann.,262(1983), 313–342. 30

work page 1983

[3] [3]

Bianchi, L

F. Bianchi, L. Brasco, R. Ognibene, On the spectrum of sets made of cores and tubes, J. Convex Anal.,31 2024, 315–358. 7, 27

work page 2024

[4] [4]

M. Sh. Birman, M. Z. Solomjak,Spectral theory of selfadjoint operators in Hilbert space.Translated from the 1980 Russian original by S. Khrushchev and V. Peller. Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, 1987. 2, 5

work page 1980

[5] [5]

Bobkov, Non-Ljusternik–Schnirelman eigenvalues of the purep−Laplacian exist, preprint (2026), https://arxiv.org/abs/2604.011388, 19, 28

V. Bobkov, Non-Ljusternik–Schnirelman eigenvalues of the purep−Laplacian exist, preprint (2026), https://arxiv.org/abs/2604.011388, 19, 28

work page arXiv 2026

[6] [6]

J. L. Bona, D. K. Bose, R. E. L. Turner, Finite-amplitude steady waves in stratified fluids, J. Math. Pures Appl. (9),62(1983), 389–439. 30

work page 1983

[7] [7]

Brasco,Handbook of Calculus of Variations for absolute beginners.Unitext,163, La Mat

L. Brasco,Handbook of Calculus of Variations for absolute beginners.Unitext,163, La Mat. per il 3+2, Springer, Cham, 2025. 8, 11, 20

work page 2025

[8] [8]

Brasco, On principal frequencies and isoperimetric ratios in convex sets, Ann

L. Brasco, On principal frequencies and isoperimetric ratios in convex sets, Ann. Fac. Sci. Toulouse Math. (6), 29(2020), 977–1005. 19

work page 2020

[9] [9]

Brasco, L

L. Brasco, L. Briani, F. Prinari, Low eigenvalues of thep−Laplacian in general open sets, preprint (2026), https://arxiv.org/abs/2602.211184, 5, 6, 13

work page arXiv 2026

[10] [10]

Brasco, L

L. Brasco, L. Briani, F. Prinari, Extremals for sharp Poincar´ e-Sobolev inequalities: periodically perforated sets and beyond, preprint (2025),https://arxiv.org/abs/2511.202606, 7, 9, 31

work page arXiv 2025

[11] [11]

Brasco, L

L. Brasco, L. Briani, F. Prinari, Extremals for Poincar´ e-Sobolev sharp constants in Steiner symmetric sets, Nonlinear Anal.,269(2026), Paper No. 114098. 6, 30

work page 2026

[12] [12]

Cuesta, D

M. Cuesta, D. Figueiredo, J.-P. Gossez, The beginning of the Fuˇ cik spectrum for thep−Laplacian, J. Differential Equations,159(1999), 212–238. 3

work page 1999

[13] [13]

DiBenedetto,C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal.,7 (1983), 827–850

E. DiBenedetto,C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal.,7 (1983), 827–850. 19

work page 1983

[14] [14]

M. J. Esteban, P.-L. Lions, Existence and nonexistence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A,93(1982/83), 1–14. 7, 22

work page 1982

[15] [15]

Franzina,Existence, Uniqueness, Optimization and Stability for low Eigenvalues of some Nonlinear Oper- ators, PhD Thesis (2012), available athttps://cvgmt.sns.it/paper/2102/3

G. Franzina,Existence, Uniqueness, Optimization and Stability for low Eigenvalues of some Nonlinear Oper- ators, PhD Thesis (2012), available athttps://cvgmt.sns.it/paper/2102/3

work page 2012

[16] [16]

Friedlander, Asymptotic behavior of the eigenvalues of thep−Laplacian, Comm

L. Friedlander, Asymptotic behavior of the eigenvalues of thep−Laplacian, Comm. Partial Differential Equa- tions,14(1989), 1059–1069. 3

work page 1989

[17] [17]

J. P. Garc´ ıa Azorero, I. Peral Alonso, Existence and nonuniqueness for thep−Laplacian: nonlinear eigenvalues, Comm. Partial Differential Equations,12(1987), 1389–1430. 3

work page 1987

[18] [18]

Guedda, L

M. Guedda, L. V´ eron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal.,13 (1989), 879–902. 23 ESSENTIAL SPECTRUM FOR THEp−LAPLACIAN 35

work page 1989

[19] [19]

P. D. Hislop, I. M. Sigal,Introduction to spectral theory. With applications to Schr¨ odinger operators, Appl. Math. Sci.,113. Springer-Verlag, New York, 1996 2

work page 1996

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Kavian, Introduction ` a la th´ eorie des points critiques et applications aux probl` emes elliptiques

O. Kavian, Introduction ` a la th´ eorie des points critiques et applications aux probl` emes elliptiques. (French) [Introduction to critical point theory and applications to elliptic problems] Math´ ematiques & Applications (Berlin) [Mathematics & Applications],13. Springer-Verlag, Paris, 1993. 26

work page 1993

[21] [21]

H. Koch, D. Tataru, Carleman estimates and absence of embedded eigenvalues, Comm. Math. Phys.,267 (2006), 419–449. 7

work page 2006

[22] [22]

E. H. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains, Invent. Math.,74 (1983), 441–448. 3

work page 1983

[23] [23]

Lˆ e, Eigenvalue problems for thep−Laplacian, Nonlinear Anal.,64(2006), 1057–1099

A. Lˆ e, Eigenvalue problems for thep−Laplacian, Nonlinear Anal.,64(2006), 1057–1099. 3

work page 2006

[24] [24]

Lindqvist, A nonlinear eigenvalue problem, inTopics in mathematical analysis, 175–203

P. Lindqvist, A nonlinear eigenvalue problem, inTopics in mathematical analysis, 175–203. Ser. Anal. Appl. Comput.,3. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. 3

work page 2008

[25] [25]

Maz’ya,Sobolev spaces with applications to elliptic partial differential equations

V. Maz’ya,Sobolev spaces with applications to elliptic partial differential equations. Second, revised and aug- mented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],342. Springer, Heidelberg, 2011. 2

work page 2011

[26] [26]

Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schr¨ odinger operator, Math

A. Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schr¨ odinger operator, Math. Scand.,8(1960), 143–153. 2, 6

work page 1960

[27] [27]

Rellich, ¨Uber das asymptotische Verhalten der L¨ osungen von ∆u+λu= 0 in unendlichen Gebieten, Jber

F. Rellich, ¨Uber das asymptotische Verhalten der L¨ osungen von ∆u+λu= 0 in unendlichen Gebieten, Jber. Deutsch. Math.-Verein.,53(1943), 57–65. 7, 22

work page 1943

[28] [28]

S. N. Roze, The spectrum of a second order elliptic operator, Mat. Sb.,80 (122)(1969), 195–209. 7

work page 1969

[29] [29]

Struwe,Variational methods

M. Struwe,Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics,34. Springer-Verlag, Berlin, 2008. 2

work page 2008

[30] [30]

Szulkin, Ljusternik-Schnirelmann theory onC 1-manifolds, Ann

A. Szulkin, Ljusternik-Schnirelmann theory onC 1-manifolds, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire,5 (1988), 119–139. 3

work page 1988

[31] [31]

Teschl,Mathematical methods in quantum mechanics

G. Teschl,Mathematical methods in quantum mechanics. With applications to Schr¨ odinger operators.Second edition. Graduate Studies in Mathematics,157. American Mathematical Society, Providence, RI, 2014. 2, 6

work page 2014

[32] [32]

K. J. Witsch, Examples of embedded eigenvalues for the Dirichlet-Laplacian in domains with infinite boundaries, Math. Methods Appl. Sci.,12(1990), 177–182. 5, 7 (L. Brasco)Dipartimento di Matematica e Informatica Universit`a degli Studi di Ferrara Via Machiavelli 35, 44121 Ferrara, Italy Email address:lorenzo.brasco@unife.it (L. Briani)School of Computati...

work page 1990