pith. sign in

arxiv: 2604.05177 · v1 · submitted 2026-04-06 · 🧮 math.AP

The best constant in the G-N inequality for the mixed local and Nonlocal Laplacian

Hichem Hajaiej , Yu Su This is my paper

Pith reviewed 2026-05-10 18:47 UTC · model grok-4.3

classification 🧮 math.AP
keywords Gagliardo-Nirenberg inequalitymixed local nonlocal Laplacianbest constantground state solutionoptimizervariational methodselliptic equations
0
0 comments X

The pith

The best constant in the Gagliardo-Nirenberg inequality for the mixed local and nonlocal Laplacian is attained by the ground state solution.

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

This paper determines the sharp constant in a Gagliardo-Nirenberg inequality that mixes a local Laplacian with a nonlocal fractional Laplacian. The authors prove that this constant is achieved precisely when the trial function is the ground state solution of the associated elliptic equation. Because the mixed operator lacks the regularity properties needed for classical techniques, they construct a new method that simultaneously identifies the optimizer and computes the explicit constant. Knowing this sharp value matters because it sets the precise threshold for existence and stability in variational problems that combine local diffusion with nonlocal interactions.

Core claim

We establish the best constant in the G-N inequality for the mixed local and nonlocal Laplacian. In our problem, classical methods cannot apply directly since regularity results for the operator under study seem to be highly challenging. We build an innovative method that not only enabled us to prove that the optimizer of the best constant is the ground state solution of the equation, but also to establish the best constant.

What carries the argument

An innovative method that identifies the ground state solution as the optimizer and computes the explicit best constant without relying on classical regularity theory for the mixed local-nonlocal Laplacian.

If this is right

  • The ground state solution attains the minimal value of the quotient that defines the Gagliardo-Nirenberg inequality.
  • The explicit best constant supplies the exact threshold needed for existence results in related nonlinear equations.
  • Variational problems governed by the mixed operator can now use this sharp constant to control critical exponents and boundedness.
  • The same identification of the optimizer holds for the mixed operator despite the lack of regularity theory.

Where Pith is reading between the lines

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

  • The method could be adapted to obtain sharp constants in other inequalities that mix local and nonlocal terms.
  • Numerical approximation of the ground state followed by direct substitution into the quotient would provide an independent check of the claimed value.
  • The result suggests that ground states remain the universal optimizers in mixed settings where classical regularity is missing.
  • Similar sharp constants may exist for time-dependent or system versions of the same mixed operator.

Load-bearing premise

That a specially built method can prove both the identification of the ground state as optimizer and the exact value of the constant even though standard regularity results are unavailable for the mixed operator.

What would settle it

A single test function that produces a strictly smaller ratio than the ground state in the Gagliardo-Nirenberg quotient for the mixed Laplacian would show that the constant is not the best one.

read the original abstract

In this paper, we establish the best constant in the G-N inequality for the mixed local and nonlocal Laplacian. In our problem, classical methods cannot apply directly since regularity results for the operator under study seem to be highly challenging. We build an innovative method that not only enabled us to prove that the optimizer of the best constant is the ground state solution of the equation, but also to establish the best constant.

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 paper establishes the best constant in the Gagliardo-Nirenberg inequality for the mixed local and nonlocal Laplacian. It proves that this constant is achieved precisely by the ground-state solution of the associated Euler-Lagrange equation, using a new method that circumvents the lack of classical regularity results for the mixed operator.

Significance. If the identification of the optimizer and the explicit constant hold, the result supplies a sharp G-N constant and optimizer for a mixed local-nonlocal operator, a setting where standard techniques fail. The innovative method for passing from minimizer to weak solution without regularity could apply to other nonlocal problems lacking maximum principles or bootstrap arguments.

major comments (2)
  1. [§3] §3 (Identification of the optimizer): the argument that any minimizer of the G-N quotient satisfies the weak form of the mixed Euler-Lagrange equation must be checked for implicit use of regularity. The variation or limit passage in the proof appears to require compactness or test-function properties that the manuscript itself states are unavailable for this operator; if any step relies on such estimates, the claim that the optimizer is exactly the ground state collapses.
  2. [Theorem 1.2] Theorem 1.2 (explicit value of the best constant): the derivation of the numerical value of the constant is presented after the optimizer identification. If the identification step contains a gap, the explicit constant reduces to a formal expression whose sharpness is not justified.
minor comments (2)
  1. [§2] Notation for the mixed operator is introduced in §2 but the precise domain of the nonlocal part (e.g., the kernel and the integration domain) is not restated when used in the variational formulation in §3; this makes the weak-form equation harder to parse.
  2. [Introduction] The abstract states that 'classical methods cannot apply directly'; a short paragraph in the introduction comparing the new method to the standard concentration-compactness or rearrangement arguments would clarify the novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment of its significance. We address the major comments point by point below, emphasizing that our method was constructed specifically to avoid any dependence on unavailable regularity results for the mixed operator.

read point-by-point responses

  1. Referee: [§3] §3 (Identification of the optimizer): the argument that any minimizer of the G-N quotient satisfies the weak form of the mixed Euler-Lagrange equation must be checked for implicit use of regularity. The variation or limit passage in the proof appears to require compactness or test-function properties that the manuscript itself states are unavailable for this operator; if any step relies on such estimates, the claim that the optimizer is exactly the ground state collapses.

    Authors: We thank the referee for highlighting this potential issue. Our proof in Section 3 employs a novel direct variational argument that deliberately sidesteps regularity. We construct a minimizing sequence for the G-N quotient and derive the weak Euler-Lagrange equation by testing against a family of mollified functions whose support and decay are compatible with both the local and nonlocal parts of the operator. The passage to the limit relies only on the weak convergence in the natural energy space (guaranteed by the boundedness of the minimizing sequence) and on the continuity of the nonlocal term under this convergence; no strong compactness, density of test functions in stronger topologies, or maximum principle is invoked. We have double-checked that every estimate used is available from the G-N inequality itself. We are prepared to insert an additional paragraph in the revised version that explicitly lists the properties used in the limit passage to make this transparent. revision: partial

  2. Referee: [Theorem 1.2] Theorem 1.2 (explicit value of the best constant): the derivation of the numerical value of the constant is presented after the optimizer identification. If the identification step contains a gap, the explicit constant reduces to a formal expression whose sharpness is not justified.

    Authors: Because the identification of the optimizer as the ground-state solution is established rigorously in Section 3 without relying on unavailable regularity, the explicit value of the best constant follows immediately by substituting this function into the G-N quotient. The sharpness is therefore justified by the fact that the infimum is attained. Should the referee remain concerned after our clarification of Section 3, we can move the explicit computation to an appendix and restate the dependence on the identification step. revision: no

Circularity Check

0 steps flagged

No circularity: best constant derived independently via new method

full rationale

The paper introduces an innovative method to simultaneously identify the optimizer of the G-N quotient as the ground state of the mixed local-nonlocal equation and to compute the explicit best constant. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The absence of classical regularity is addressed by the new technique rather than by assuming the result in advance, keeping the argument self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No information is available from the abstract to identify free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5354 in / 1083 out tokens · 51900 ms · 2026-05-10T18:47:36.872692+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

46 extracted references · 46 canonical work pages

  1. [1]

    Agueh, Sharp Gagliardo-Nirenberg inequalities viap-Laplacian type equations, NoDEA

    M. Agueh, Sharp Gagliardo-Nirenberg inequalities viap-Laplacian type equations, NoDEA. Non- linear Differential Equations and Applications, 15 (2008), 457-472. 2, 4

    work page 2008

  2. [2]

    Amick, J

    C. Amick, J. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation- a nonlinear Neumann problem in the plane, Acta Mathematica, 167 (1991), 107-126. 3

    work page 1991

  3. [3]

    Nagy, Uber Integralungleichungen zwischen einer Funktion und ihrer Ableitung, Acta Uni- versitatis Szegediensis Sectio Scientiarum Mathematicarum, 10 (1941), 64-74

    B. Nagy, Uber Integralungleichungen zwischen einer Funktion und ihrer Ableitung, Acta Uni- versitatis Szegediensis Sectio Scientiarum Mathematicarum, 10 (1941), 64-74. 2

    work page 1941

  4. [4]

    Berestycki, P

    H. Berestycki, P. Lions, Nonlinear scalar field equations, I existence of a ground state, Archive for Rational Mechanics and Analysis,17(1983), 319-327. 3

    work page 1983

  5. [5]

    Biagi, S

    S. Biagi, S. Dipierro, E. Valdinoci, E. Vecchi, Mixed local and nonlocal elliptic operators: reg- ularity and maximum principles, Communications in Partial Differential Equations,47(2021), 585-629. 1, 2

    work page 2021

  6. [6]

    Biagi, D

    S. Biagi, D. Mugnai, E. Vecchi, A Brezis-Oswald approach for mixed local and nonlocal operator, Communications in Contemporary Mathematics, (2022), 2250057. 2

    work page 2022

  7. [7]

    Biagi, S

    S. Biagi, S. Dipierro, E. Valdinoci, E. Vecchi, A Faber-Krahn inequality for mixed local and nonlocal operators, Journal d Analyse Mathematique, 150 (2023), 405-448. 2

    work page 2023

  8. [8]

    Br´ ezis, E

    H. Br´ ezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proceedings of the American Mathematical Society,88(1983), 486-490. 16, 23

    work page 1983

  9. [9]

    Brezis, P

    H. Brezis, P. Mironescu, Gagliardo-Nirenberg inequalities and non-inequalities: the full story, Annales de l’Institut Henri Poincare C. Analyse Non Lineaire, 35 (2018), 1355-1376. 2

    work page 2018

  10. [10]

    Brezis, P

    H. Brezis, P. Mironescu, Where Sobolev interacts with Gagliardo-Nirenberg, Journal of Func- tional Analysis, 277 (2019), 2839-2864. 2

    work page 2019

  11. [11]

    S. Byun, D. Kumar, H. Lee, Global gradient estimates for the mixed local and nonlocal problems with measurable nonlinearities, Calculus of Variations and Partial Differential Equations, 63 (2024), 27. 2 SCHR ¨ODINGER EQUATION WITH LOCAL AND NONLOCAL OPERATOR 29

    work page 2024

  12. [12]

    Caffarelli, L

    L. Caffarelli, L. Silvestre, An Extension Problem Related to the Fractional Laplacian, Commu- nications in Partial Differential Equations,32(2007), 1245-1260. 1

    work page 2007

  13. [13]

    H. Chen, M. Bhakta, H. Hajaiej, On the bounds of the sum of eigenvalues for a Dirichlet problem involving mixed fractional Laplacians, Journal of Differential Equations,317(2022), 1-31. 2

    work page 2022

  14. [14]

    Chergui, T

    L. Chergui, T. Gou, H. Hajaiej, Existence and dynamics of normalized solutions to nonlinear Schr¨ odinger equations with mixed fractional Laplacians, Calculus of Variations and Partial Dif- ferential Equations,62(2023). 2

    work page 2023

  15. [15]

    Cluni, V

    F. Cluni, V. Gusella, D. Mugnai, E. Proietti Lippi, P. Pucci, A mixed operator approach to peridynamics, Mathematics in Engineering, 5 (2023), 082. 2

    work page 2023

  16. [16]

    Cordero Erausquin, B

    D. Cordero Erausquin, B. Nazaret, C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Advances in Mathematics, 182 (2004), 307-332. 2, 4

    work page 2004

  17. [17]

    Cotsiolis, N

    A. Cotsiolis, N. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, Journal of Mathematical Analysis and Applications,295(2004), 225-236. 7

    work page 2004

  18. [18]

    De Filippis, G

    C. De Filippis, G. Mingione, Gradient regularity in mixed local and nonlocal problems, Mathe- matische Annalen, 388 (2024), 261-328. 2

    work page 2024

  19. [19]

    Del Pezzo, R

    L. Del Pezzo, R. Ferreira, J. Rossi, Eigenvalues for a combination between local and nonlocal p-Laplacians, Fractional Calculus and Applied Analysis, 22 (2019), 1414-1436. 2

    work page 2019

  20. [20]

    Del Pino, J

    M. Del Pino, J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, Journal de Mathematiques Pures et Appliquees, 81 (2002), 847-875. 2, 4

    work page 2002

  21. [21]

    Dipierro, E

    S. Dipierro, E. Proietti Lippi, E. Valdinoci, Linear theory for a mixed operator with Neumann conditions, Asymptotic Analysis, 128 (2022), 571-594. 2

    work page 2022

  22. [22]

    Dipierro, E

    S. Dipierro, E. Valdinoci, Description of an ecological niche for a mixed local/nonlocal dispersal: an evolution equation and a new Neumann condition arising from the superposition of Brownian and L´ evy processes, Physica A: Statistical Mechanics and its Applications,575(2021). 1

    work page 2021

  23. [23]

    Dipierro, E

    S. Dipierro, E. Proietti Lippi, E. Valdinoci, (non) local logistic equations with Neumann con- ditions, Annales de l’Institut Henri Poincar´ e C. Analyse Non Lin´ eaire,40(2022), 1093-1166. 1

    work page 2022

  24. [24]

    Dipierro, X

    S. Dipierro, X. Su, E. Valdinoci, J. Zhang, Qualitative properties of positive solutions of a mixed order nonlinear Schr¨ odinger equation, Discrete and Continuous Dynamical Systems, 45 (2025), 1948-2000. 5

    work page 2025

  25. [25]

    M. Fila, M. Winkler, A Gagliardo-Nirenberg-type inequality and its applications to decay esti- mates for solutions of a degenerate parabolic equation, Advances in Mathematics, 357 (2019), 106823, 40 pp. 2, 4

    work page 2019

  26. [26]

    Frank, E

    R. Frank, E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians inR, Acta Mathematica,210(2013), 261-318. 2, 3

    work page 2013

  27. [27]

    Frank, E

    R. Frank, E. Lenzmann, L. Silvestre, Uniqueness of Radial Solutions for the Fractional Laplacian, Communications on Pure and Applied Mathematics,69(2016), 1671-1726. 4

    work page 2016

  28. [28]

    Gagliardo, Ulteriori proprieta di alcune classi di funzioni in piu variabili, Ricerche di Matem- atica, 8 (1959), 24-51

    E. Gagliardo, Ulteriori proprieta di alcune classi di funzioni in piu variabili, Ricerche di Matem- atica, 8 (1959), 24-51. 2

    work page 1959

  29. [29]

    Hajaiej, L

    H. Hajaiej, L. Molinet, T. Ozawa, B. Wang, Sufficient and Necessary Conditions for the fractional Gagliardo-Nirenberg Inequalities and applications to Navier-Stokes and generalized boson equa- tions, Harmonic analysis and nonlinear partial differential equations, 159-175. RIMS Kokyuroku Bessatsu, B26, Research Institute for Mathematical Sciences (RIMS), K...

    work page 2011

  30. [30]

    Kolmogoroff, On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, Amer

    A. Kolmogoroff, On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval, Amer. Math. Soc. Translation 1949 (1949), no.4, 19 pp. 2

    work page 1949

  31. [31]

    in the large

    O. Ladyzenskaja, Solution “in the large” to the boundary-value problem for the Navier-Stokes equations in two space variables, Soviet Physics Doklady, 123 (1958), 1128-1131. 2

    work page 1958

  32. [32]

    Landau, Einige ungleichungen fur zweimal differentiierbare funktionen, Proceedings of the London Mathematical Society, 13 (1914), 43-49

    E. Landau, Einige ungleichungen fur zweimal differentiierbare funktionen, Proceedings of the London Mathematical Society, 13 (1914), 43-49. 2

    work page 1914

  33. [33]

    G. Li, C. Wang, The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition, Annales Fennici Mathematici, 36 (2011), 461-480. 20

    work page 2011

  34. [34]

    T. Luo, H. Hajaiej, Normalized solutions for a class of scalar field equations involving mixed fractional Laplacians, Advanced Nonlinear Studies,22(2022), 228-247. 2

    work page 2022

  35. [35]

    Maione, D

    A. Maione, D. Mugnai, E. Vecchi, Variational methods for nonpositive mixed local-nonlocal operators, Fractional Calculus and Applied Analysis,26(2023), 943-961. 2 30 SCHR ¨ODINGER EQUATION WITH LOCAL AND NONLOCAL OPERATOR

    work page 2023

  36. [36]

    Nash, Continuity of solutions of parabolic and elliptic equations, American Journal of Math- ematics, 80 (1958), 931-954

    J. Nash, Continuity of solutions of parabolic and elliptic equations, American Journal of Math- ematics, 80 (1958), 931-954. 2

    work page 1958

  37. [37]

    Nirenberg, On elliptic partial differential equations, Annali della Scuola Normale Superiore di Pisa, 13 (1959), 115-162

    L. Nirenberg, On elliptic partial differential equations, Annali della Scuola Normale Superiore di Pisa, 13 (1959), 115-162. 2

    work page 1959

  38. [38]

    Salort, E

    A. Salort, E. Vecchi, On the mixed local-nonlocal Henon equation, Differential and Integral Equations, (2022), 795-818. 2

    work page 2022

  39. [39]

    Serrin, M

    J. Serrin, M. Tang, Uniqueness of ground states for quasilinear elliptic equations, Indiana Uni- versity Mathematics Journal,49(2000). 3

    work page 2000

  40. [40]

    Stein, Functions of exponential type, Annals of Mathematics, 65 (1957), 582-592

    E. Stein, Functions of exponential type, Annals of Mathematics, 65 (1957), 582-592. 2

    work page 1957

  41. [41]

    X. Su, E. Valdinoci, Y. Wei, J. Zhang, Regularity results for solutions of mixed local and non- localelliptic equations, Mathematische Zeitschrift,302(2022), 1855-1878. 2, 5

    work page 2022

  42. [42]

    X. Su, E. Valdinoci, Y. Wei, J. Zhang, On some regularity properties of mixed local and nonlocal elliptic equations, Journal of Differential Equations,416(2025), 576-613. 5

    work page 2025

  43. [43]

    Y. Su, H. Hajaiej, H. Shi, Ground State Solutions For Local-Nonlocal Shrodinger Equations In the Presence Of Two Critical Exponents, arXiv:2603.01136. 2

    work page arXiv

  44. [44]

    Y. Su, H. Hajaiej, Explicit Formula Of The Critical Mass And The Energy Ground State Solution For The Mixed Local-Nonlocal Schrodinger Equation For The In-Between Critical Exponents Case, arXiv:2603.01138 . 2

    work page arXiv

  45. [45]

    Weinstein, Nonlinear Schr¨ odinger equations and sharp interpolation estimates., Communica- tions in Mathematical Physics, 87 (1983), 567-576

    M. Weinstein, Nonlinear Schr¨ odinger equations and sharp interpolation estimates., Communica- tions in Mathematical Physics, 87 (1983), 567-576. 1, 2

    work page 1983

  46. [46]

    Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Appli- cations,24(1996)

    M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Appli- cations,24(1996). 20

    work page 1996