Asymptotics of Dirichlet Problems to Fractional p-Laplacian Functionals-Approach in De Giorgi Sense

Raphael Feng Li

arxiv: 1907.08028 · v1 · pith:E66Q422Wnew · submitted 2019-07-18 · 🧮 math.AP

Asymptotics of Dirichlet Problems to Fractional p-Laplacian Functionals-Approach in De Giorgi Sense

Raphael Feng Li This is my paper

Pith reviewed 2026-05-24 19:44 UTC · model grok-4.3

classification 🧮 math.AP

keywords fractional p-LaplacianGamma-convergenceinfinity-LaplacianDe Giorgi senseDirichlet boundary conditionasymptoticsHolder infinity-Laplacianfractional Sobolev norms

0 comments

The pith

As p tends to infinity, minimizers of fractional p-Laplacians with fixed Dirichlet data Gamma-converge to minimizers of the Holder infinity-Laplacian.

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

The paper establishes Gamma-convergence results for fractional p-Laplacian functionals in the De Giorgi sense. It shows that the non-homogeneous Dirichlet problem for these functionals converges, as p goes to infinity, to the corresponding problem for the Holder infinity-Laplacian. Separate asymptotic analyses are given for the fractional order parameter k approaching s from above in the non-homogeneous case and from below in the free case, revealing distinct behaviors in the latter. These limits connect sequences of minimizers at finite p to the limiting infinity problem under the same boundary conditions. A reader would care because the results supply a rigorous passage from tractable p-approximations to the infinity-Laplacian limit.

Core claim

The paper proves that as p→+∞ the minimizers of the fractional p-Laplacian with Dirichlet boundary Gamma-converge to a minimizer of the Holder infinity-Laplacian under the same Dirichlet boundary condition. The analysis is performed entirely in the De Giorgi sense for both the p-to-infinity limit and the two directions of approach of the fractional parameter k to s.

What carries the argument

Gamma-convergence of the fractional W^{s,p}-norm functionals in the De Giorgi sense, which links the Dirichlet problems at finite p to the infinity-Laplacian limit.

If this is right

Minimizers of the fractional p-Dirichlet problem converge to minimizers of the infinity-Laplacian Dirichlet problem.
The same Gamma-convergence holds for the non-homogeneous fractional p-functionals when the fractional order approaches its target from above.
Approaching the fractional order from below in the free (no-boundary) case produces a different limiting behavior.
Both the functionals and their minimizers are controlled by the De Giorgi Gamma-limit throughout the approximation processes.

Where Pith is reading between the lines

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

Numerical schemes that solve large-p fractional problems could approximate infinity-Laplacian solutions without directly discretizing the infinity operator.
The observed difference between approaching the fractional order from above and from below may indicate that boundary conditions interact asymmetrically with the nonlocal tail.
The De Giorgi framework used here could be tested on other nonlocal operators whose p-versions are known to admit Gamma-limits.
Regularity or uniqueness properties known for the infinity-Laplacian might transfer back to uniform estimates on the p-approximations.

Load-bearing premise

The Gamma-convergence statements require that the entire analysis be performed in the De Giorgi sense.

What would settle it

An explicit sequence of functions whose fractional p-energies fail to approach the infinity-Laplacian energy under the same Dirichlet data as p tends to infinity would falsify the convergence claim.

read the original abstract

In this paper we firstly study the limit of minimizers of the fractional $W^{s,p}$-norms as $p\rightarrow+\infty$ in De Giorgi sense. In particular, we analyzed the $\Gamma$-convergence of non-homogeneous Dirichlet boundary problem for fractional $p$-Laplacian in this approximation process, and proved that as $p\rightarrow+\infty$ the minimizers of fractional $p$-Laplacian with Dirichlet boundary $\Gamma$-converges to a minimizer of H\"{o}lder $\infty$-Laplacian under the same Dirichlet boundary condition. On the other hand, we first investigate the asymptotic behaviour of non-homogeneous fractional $p$-functionals when $k\rightarrow s$ from above; then we study the approximation process as $k\rightarrow s$ from below of a free fractional $p$-functional, during which we will find some special phenomenon different from the case from above. Both of the way to dispose these two asymptotic directions are in the De Giorgi sense.

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's main claim is a Gamma-convergence result for non-homogeneous Dirichlet fractional p-Laplacians to a Holder infinity-Laplacian as p to infinity, plus some k to s limits, all in De Giorgi sense.

read the letter

The central result here is the Gamma-convergence of minimizers for the non-homogeneous fractional p-Laplacian Dirichlet problem to a minimizer of the Holder infinity-Laplacian as p goes to infinity, along with asymptotic analysis of the functionals as k approaches s from above and from below. The from-below case for the free functional is presented as showing distinct behavior. These directions are stated as new. The work applies standard Gamma-convergence methods in the De Giorgi framework, which is a natural fit for connecting the variational problems across these limits, and it keeps the boundary conditions consistent in the statements. That gives a clear, concrete set of claims about the convergence of minimizers. The main limitation is that the abstract gives no derivations or error estimates, so the technical details of how the non-homogeneous terms and the infinity-Laplacian limit are handled would need checking in the full proofs. The De Giorgi approach itself is appropriate and not a stretch, but any gaps in the estimates for the specific regimes could affect how solid the convergence statements turn out to be. The overall scope stays narrow, inside fractional Sobolev spaces and these particular variational limits. This is for readers already working on p-Laplacian asymptotics or infinity-Laplace equations in variational settings; a specialist would pick up the new regimes and the from-below observation. It deserves peer review because the claims are explicit, the methods are established in the area, and the results add targeted information even if the proofs require scrutiny.

Referee Report

0 major / 2 minor

Summary. The paper studies asymptotic limits of Dirichlet problems for fractional p-Laplacian functionals in the De Giorgi sense. It claims to prove that, as p→+∞, minimizers of the non-homogeneous fractional p-Laplacian Gamma-converge to a minimizer of the Hölder ∞-Laplacian under the same Dirichlet boundary condition. It further examines the asymptotic behavior of non-homogeneous fractional p-functionals as k→s from above, and the approximation of a free fractional p-functional as k→s from below, again in the De Giorgi sense, noting phenomena that differ between the two directions.

Significance. If the Gamma-convergence statements hold, the results would supply a variational framework linking fractional p-Laplacians to infinity-Laplacians via De Giorgi convergence, which is relevant for understanding limiting regimes in nonlocal variational problems and could inform regularity theory or approximation schemes.

minor comments (2)

The abstract contains minor grammatical issues (e.g., 'we firstly study', 'Hölder ∞-Laplacian') that should be polished for clarity.
Notation for the fractional order (s vs. k) and the precise definition of the Hölder ∞-Laplacian functional should be introduced consistently in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of our work on the Gamma-convergence of fractional p-Laplacian Dirichlet problems to the Holder infinity-Laplacian and the related asymptotic analyses in the De Giorgi sense. The report does not list any specific major comments, so we provide no point-by-point responses below. We note the 'uncertain' recommendation and remain available to clarify any aspects of the proofs if requested.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained variational analysis

full rationale

The paper's central result is a Γ-convergence statement (in the De Giorgi sense) for fractional p-Laplacian functionals with Dirichlet data as p→∞ to a Hölder ∞-Laplacian limit, plus related k↔s asymptotics. These are established by direct comparison of energy functionals and minimizers under the stated boundary conditions; no step reduces a claimed prediction or limit to a fitted parameter, self-defined quantity, or load-bearing self-citation. The abstract and described argument invoke standard variational tools without importing uniqueness theorems or ansatzes from prior author work. This matches the default expectation of a non-circular paper whose claims rest on independent analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background results from functional analysis and the theory of Gamma-convergence; no new free parameters or postulated entities are introduced.

axioms (1)

standard math Standard properties of fractional Sobolev spaces W^{s,p} and the definition of Gamma-convergence in the De Giorgi sense
Invoked to justify passage to the limit for minimizers of the fractional p-norms.

pith-pipeline@v0.9.0 · 5709 in / 1228 out tokens · 28782 ms · 2026-05-24T19:44:24.062045+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.

proved that as p→+∞ the minimizers of fractional p-Laplacian with Dirichlet boundary Γ-converges to a minimizer of Hölder ∞-Laplacian
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

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

Γ-convergence of non-homogeneous Dirichlet boundary problem for fractional p-Laplacian

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.

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Works this paper leans on

46 extracted references · 46 canonical work pages · 1 internal anchor

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Franzina, G., Palatucci, G., Fractional p-eigenvalue s, Riv. Mat. Univ. Parma, 5, 315-328 (2014)

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Dyn., 5, 385-424 (2005)

Guan, Q.Y., Ma, Z.M., Boundary Problems for Fractional Laplacians, Stoch. Dyn., 5, 385-424 (2005)

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Ishii, H., Nakamura, G., A class of integral equations a nd approximation of p-Laplace equa- tions, Calc. Var. 37, 485-522, (2010)

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[1] [1]

134, 377-403 (2011)

Ambrosio, L., De Philippis, G., Martinazzi, L., Gamma-c onvergence of nonlocal perimeter functionals, manuscripta math. 134, 377-403 (2011)

work page 2011

[2] [2]

Aronsson, G., Minimization problems for the functional supxF (x, f(x), f ′(x)), Ark. Mat. 6, 33-53 (1965)

work page 1965

[3] [3]

II , Ark

Aronsson, G., Minimization problems for the functional supxF (x, f(x), f ′(x)). II , Ark. Mat. 6, 409-431 (1966)

work page 1966

[4] [4]

Aronsson, G., Extention of functions satisfying Lipsch itz conditions, Ark. Mat. 6, 551-561 (1967)

work page 1967

[5] [5]

Aronsson, G., On the partial diﬀerential equation u2 xuxx + 2uxuyuxy + u2 yuyy = 0, Ark. Mat. 7, 3954251 (1968)

work page 1968

[6] [6]

III , Ark

Aronsson, G., Minimization problems for the functional supxF (x, f(x), f ′(x)). III , Ark. Mat. 7, 509-512 (1969)

work page 1969

[7] [7]

Aronsson, G., Crandall, M.G., Juutinen, P., A tour of the theory of absolutely minimizing functions, Bull. Amer. Math. Soc. (N.S.) 41, 439-505 (2004)

work page 2004

[8] [8]

Nonlinear Analysis, Diﬀerential Equa- tions and Control (Clarke and Stern eds.)

Barron, E.N., Viscosity solutions and analysis in L∞ . Nonlinear Analysis, Diﬀerential Equa- tions and Control (Clarke and Stern eds.). Kluwer, Dordrech t, (1999)

work page 1999

[9] [9]

Bogdan, K., Byczkowski, T., Potential theory for the α-stable Schr¨ odinger operator on bounded Lipschitz domains, Studia Math., 133, 53-92 (1999)

work page 1999

[10] [10]

Bjorland, C., Caﬀarelli, L.A., and Figalli, A., Non-Lo cal Tug-of-W ar and the Inﬁnity Frac- tional Laplacian. Comm. Pure. Appl. Math. (2011)

work page 2011

[11] [11]

Bhattacharya, T., Di Benedetto, E., and Manfredi, J., L imites as p →∞ of ∆ pup = f and related extremal problems. Rend. Sem. Mat. Univ. Politec. T orino (1989), No, Special Issue 15-68 (1991), Some topics in nonlinear PDEs, Turin (1989)

work page 1989

[12] [12]

der Math

Brasco, L., Franzina, G., A note on positive eigenfunct ions and hidden convexity, Arch. der Math. 99, 367-374 (Oct. 2012)

work page 2012

[13] [13]

An analyt ic and Probabilistic Approach to Balayage

Bliedtner, J., Hansen, W., Potential Theory. An analyt ic and Probabilistic Approach to Balayage. Univertext. Springer-Verlag, Berlin (1986)

work page 1986

[14] [14]

Barron, E.N., Jensen, R.R., W ang, C.Y., Lower semicont inuity of L∞ functionals, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire, 18, 495-517 (2001)

work page 2001

[15] [15]

Barron, E.N., Jensen, R.R., W ang, C.Y., The Euler equat ion and absolute minimizers of L∞ functionals, Arch. Ration. Mech. Anal., 157, 255-283 (2001)

work page 2001

[16] [16]

Brasco, L., Lindgren, E., Parini, E., The fractional Ch eeger problem, Interfaces Free Bound., 16, 419-458 (2014)

work page 2014

[17] [17]

73 NO.1 , 110-121 (2010)

Bocea, M., Mih˘ ailescu, M., Γ-convergence of power-la w functionals with variable exponents, Nonlinear Anal. 73 NO.1 , 110-121 (2010)

work page 2010

[18] [18]

Bocea, M., Nesi, V., Γ-convergence of power-law functi onals, variational principles in L∞ , and applications, 39, 1550-1576, (2008) 20 RAPHAEL FENG LI

work page 2008

[19] [19]

Bonder, J.F., P´ erez-Llanos, M., Salort, A.M., A H¨ older Inﬁnity Laplacian Obtained as Limit of Orlicz Fractional Laplacians, ArXive:1807.01669v1 (20 18)

work page internal anchor Pith review Pith/arXiv arXiv

[20] [20]

Brasco, L., Parini, E., Squassina, M., STABILITY OF V AR IATIONAL EIGENV ALUES FOR THE FRACTIONAL p-LAPLACIAN, arxive:1503.0418v1 [math .AP] 13 Mar (2015)

work page arXiv 2015

[21] [21]

Oxford Univ ersity Press, Oxford, (2002)

Braides, A., Γ-convergence for Beginners. Oxford Univ ersity Press, Oxford, (2002)

work page 2002

[22] [22]

Crandall, M.G., Evans, L.C., Gariepy, R.F., Optimal Li pschitz extensions and the inﬁnity laplacian. Cal. Var. 13, 123-139 (2001)

work page 2001

[23] [23]

Chambolle, A., Lindgren, E., Monneau, R., A H¨ older inﬁnity laplacian, ESAIM: COCV 18, 799-835 (2012)

work page 2012

[24] [24]

Virkh¨ auser, Basel (1993)

Dal Maso, G., An introduction to Γ-convergence. Virkh¨ auser, Basel (1993)

work page 1993

[25] [25]

De Giorgi, E., Sulla convergenza di alcune successioni di integrali del tipo dell’area., Rend. Mat. 8, 277-294 (1975)

work page 1975

[26] [26]

Di Castro, A., Kuusi, T., Palatucci, G., Local behavior of fractional p-minimizers, Ann. I. H. Poincar´e. 33, 1279-1299 (2016)

work page 2016

[27] [27]

43, 593-609 (2015)

Degiovanni, M., Marzocchi, M., On the Dependence on p of the Variational Eigenvalues of the p-Laplace operator, Potential Anal. 43, 593-609 (2015)

work page 2015

[28] [28]

Di Nezza, E., Palatucci, G., Valdinoci, E., Hichhiker’ s guide to the fractional Sobolev spaces. Bull. Sci. Math. Soc. 136, 521-573 (2012)

work page 2012

[29] [29]

CRC Press, W ashington (1992)

Evans, L.C., Gariepy, R.F., Mesure Theory and Fine Prop erties of Functions, Studies in Advanced Mathematics. CRC Press, W ashington (1992)

work page 1992

[30] [30]

Springer Monographs in Mathematics

Fonseca, I., Leoni, G., Modern methods in the Calculus o f Variations: Lp spaces. Springer Monographs in Mathematics. Springer, New York (2007)

work page 2007

[31] [31]

Franzina, G., Palatucci, G., Fractional p-eigenvalue s, Riv. Mat. Univ. Parma, 5, 315-328 (2014)

work page 2014

[32] [32]

Dyn., 5, 385-424 (2005)

Guan, Q.Y., Ma, Z.M., Boundary Problems for Fractional Laplacians, Stoch. Dyn., 5, 385-424 (2005)

work page 2005

[33] [33]

Ishii, H., Nakamura, G., A class of integral equations a nd approximation of p-Laplace equa- tions, Calc. Var. 37, 485-522, (2010)

work page 2010

[34] [34]

Jost, J., Li-Jost, X., Calculus of Variations, Cambrid ge Studied in Advanced Mathematics, V ol.64, Cambridge University Press, Cambridge, (1998)

work page 1998

[35] [35]

Li, R.F., On Relative-nonlocal p-Rayleigh Quotients, to appear

work page

[36] [36]

Lindgren, E., H¨ older estimates for viscosity solutions of equations of frac tional p-Laplace type, Nonlinear. Diﬀer. Equ. Appl., 23:5:55 18pp, (2016)

work page 2016

[37] [37]

Lindgren, E., Lindqvist, P., Fractional eigenvalues. Calc. Var. 49, 795-826, (2014)

work page 2014

[38] [38]

Springer, Berlin (201 1)

Maz´ya, V., Sobolev Spaces, 2nd, revised and augmented Edition, G rundlehren der mathe- matischen Wisssenschaften, Vol 342. Springer, Berlin (201 1)

work page

[39] [39]

Palatucci, G., The Dirichlet problem for the p-fractio nal Laplace equation, Nonlinear Anal- ysis, 177, 699-732 (2018)

work page 2018

[40] [40]

Ponce, A.C., A new approach to Sobolev spaces and connec tions to Γ-convergence, Calc. Var. 19, 229-255 (2004)

work page 2004

[41] [41]

Ros-Oton, X., Serra, J., Fractional Laplacian: Pohoza ev identidy and nonexistence results. C. R. Math. Acad. Sci. Paris, 350, 505-508 (2012)

work page 2012

[42] [42]

Rational Mech

Ros-Oton, X., Serra, J., the Pohozaev identidy for the f ractional Laplacian, Arch. Rational Mech. Anal, 213, 587-628 (2014)

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