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arxiv: 2604.13498 · v1 · submitted 2026-04-15 · 🧮 math.AP

Homogenization and integral representation of energy functionals in manifold valued Orlicz-Sobolev spaces

Joseph Dongho , Joel Fotso Tachago , Franck Tchinda , Elvira Zappale This is my paper

Pith reviewed 2026-05-10 12:49 UTC · model grok-4.3

classification 🧮 math.AP
keywords Orlicz-Sobolev spacesGamma-convergencehomogenizationdimensional reductiontangential quasiconvexityintegral representationmanifold-valued maps
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The pith

The Gamma-limit of homogenized manifold-valued energies in Orlicz-Sobolev spaces is a tangential quasiconvex integrand given by a cell formula.

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

This paper extends classical integral representation results for the simultaneous homogenization and dimensional reduction of energies on fields taking values in a differentiable manifold to the setting of Orlicz-Sobolev spaces. It shows via Gamma-convergence that the density of the resulting limit functional is a tangential quasiconvex integrand recoverable from a cell problem. The argument relies on the Young function satisfying the Delta-two and gradient-two conditions that control growth. A parallel integral representation is established without the manifold constraint in the general Orlicz case. Readers care because these spaces accommodate growth rates beyond the power-law behavior of standard Sobolev spaces, widening the range of admissible models.

Core claim

Due to the Delta-two and gradient-two conditions verified by the Young function Phi, which modulates the growth behaviour, the density of the Gamma-limit is a tangential quasiconvex integrand represented by a cell formula. This holds for the simultaneous homogenization and dimensional reduction of integral energies defined on manifold-valued fields in Orlicz-Sobolev spaces, and a general integral representation result is proved in the unconstrained Orlicz setting.

What carries the argument

The cell formula for the tangential quasiconvex density of the Gamma-limit, which encodes the effective energy after homogenization and dimensional reduction while respecting the manifold constraint.

If this is right

  • The effective energy after simultaneous homogenization and dimensional reduction can be computed by minimizing a periodic cell problem for any admissible Orlicz growth.
  • Integral representation via tangential quasiconvexity holds for manifold-constrained variational problems in Orlicz-Sobolev spaces.
  • A general integral representation result applies to unconstrained integral functionals in the Orlicz setting.

Where Pith is reading between the lines

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

  • The cell-formula approach may extend to related Gamma-convergence problems in Orlicz spaces whenever similar modular growth conditions hold.
  • Borderline cases where the Delta-two condition fails could be tested with explicit examples to see whether alternative representations become necessary.

Load-bearing premise

The Young function Phi must satisfy the Delta-two and gradient-two conditions to guarantee the growth behavior allowing the Gamma-limit to exist and admit a cell-formula representation.

What would settle it

A concrete counterexample consisting of a Young function that violates the Delta-two or gradient-two condition together with a sequence of manifold-valued fields whose Gamma-limit energy density cannot be recovered from any cell formula.

read the original abstract

This paper aims to extend to Orlicz-Sobolev spaces some results of integral representation for the simultaneous homogenization and dimensional reduction of integral energies defined on fields taking values on a differentiable manifold. Since our functional framework goes beyond the classical Sobolev's spaces, we also prove, via $\Gamma$-convergence, a general integral representation results in the unconstrained Orlicz setting. Due to $\Delta_2$ and $\nabla_2$ conditions verified by the Young function $\Phi$ (which modulated the growth behaviour), we prove that the density of the $\Gamma$-limit is a tangential quasiconvex integrand represented by a cell formula.

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 / 2 minor

Summary. The paper extends results on simultaneous homogenization and dimensional reduction of integral energies for manifold-valued fields to Orlicz-Sobolev spaces. It also proves a general integral representation result in the unconstrained Orlicz setting via Γ-convergence. Under the Δ₂ and ∇₂ conditions on the Young function Φ, the density of the Γ-limit is shown to be a tangential quasiconvex integrand represented by a cell formula.

Significance. If the central claims hold, this provides a meaningful extension of homogenization theory to non-standard growth conditions and manifold constraints, which is useful for variational problems in materials science and geometry. The Γ-convergence framework and cell-formula representation are standard tools that, when adapted here, offer a concrete and potentially computable limit density. The verification of the Δ₂ and ∇₂ conditions on Φ is a standard but essential step that secures compactness and lower semicontinuity.

minor comments (2)
  1. The abstract and introduction should more explicitly distinguish the new contributions in the manifold-valued Orlicz case from the unconstrained Orlicz result and from prior Sobolev-space work.
  2. Notation for the Orlicz-Sobolev space W^{1,Φ} and the manifold constraint should be recalled in the preliminaries section for self-contained reading.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and for recommending minor revision. The report accurately summarizes the main contributions concerning the extension of homogenization and integral representation results to manifold-valued fields in Orlicz-Sobolev spaces under the given growth conditions. No specific major comments or points requiring clarification were raised.

Circularity Check

0 steps flagged

No significant circularity; standard Γ-convergence extension with cell formula

full rationale

The paper claims to prove, via Γ-convergence, that the density of the Γ-limit in the manifold-valued Orlicz-Sobolev setting is a tangential quasiconvex integrand given by a cell formula, relying on the standard Δ₂ and ∇₂ conditions on Φ for growth and compactness. These conditions are external to the target result and are invoked only to guarantee the applicability of existing Γ-convergence machinery and integral representation theorems (which the paper extends rather than re-derives). The cell formula is the standard definition of the homogenized density in homogenization theory, not a fitted parameter or self-referential quantity. No load-bearing self-citation chain, ansatz smuggling, or renaming of known results is indicated in the provided abstract or reader summary; the derivation remains self-contained against external variational analysis benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the Δ₂ and ∇₂ conditions on the Young function Φ and on standard background results from Gamma-convergence theory; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Young function Φ satisfies Δ₂ and ∇₂ conditions
    Explicitly invoked in the abstract to control growth and guarantee the Gamma-limit representation.
  • standard math Standard Gamma-convergence and quasiconvexity theory in Sobolev spaces extends to Orlicz-Sobolev setting under the stated growth conditions
    Used to obtain the integral representation and cell formula.

pith-pipeline@v0.9.0 · 5412 in / 1398 out tokens · 31367 ms · 2026-05-10T12:49:33.911935+00:00 · methodology

discussion (0)

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Works this paper leans on

79 extracted references · 79 canonical work pages

  1. [1]

    Adams R. A. , On the Orlicz-Sobolev imbedding theorem, J. Funct. Anal., 24, (1977), 241-257

    work page 1977

  2. [2]

    Acerbi, G

    E. Acerbi, G. Buttazzo, D. Percivale , A variational definition of the strain energy for an elastic string. Journal of Elasticity, 25, (1991), 137-148

    work page 1991

  3. [3]

    Alicandro, A

    R. Alicandro, A. Corbo Esposito, C. Leone:

    work page

  4. [4]

    R. A. Adams , Sobolev Spaces , Academic Press, New York, 1975

    work page 1975

  5. [5]

    Alberico, A

    A. Alberico, A. Cianchi , Differentiability properties of Orlicz-Sobolev functions, Ark. Mat. 43 , (2005), 1-28

    work page 2005

  6. [6]

    Alicandro, C

    R. Alicandro, C. Leone , 3D-2D asymptotic analysis for micromagnetic energies. ESAIM: COCV, 6 (2001), 489–498

    work page 2001

  7. [7]

    Allaire , Homogenization and two scale convergence, SIAM, J

    G. Allaire , Homogenization and two scale convergence, SIAM, J. Math. Anal., 23, (1992), 1482-1518

    work page 1992

  8. [8]

    Ambrosio, A

    L. Ambrosio, A. Braides: Functionals defined on partitions in sets of finite perimeter. I. Integral representation and G-convergence. Journal de Mathématiques Pures et Appliquées, 69 (1990), 285-305

    work page 1990

  9. [9]

    Alibert, G

    J.J. Alibert, G. Bouchitté : Non-uniform integrability and generalized Young measures. J. Convex

    work page

  10. [10]

    Ambrosio and G

    L. Ambrosio and G. Dal Maso , On the relaxation in BV( , R ^ m ) of quasiconvex integrals. J. Funct. Anal. 109 (1992) 76–97

    work page 1992

  11. [11]

    Ambrosio, G

    L. Ambrosio, G. Dal Maso ,

    work page

  12. [12]

    Ambrosio, N

    L. Ambrosio, N. Fusco, D. Pallara , Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York, 2000

    work page 2000

  13. [13]

    Babadjian, V

    J.-F. Babadjian, V. Millot , Homogenization of variational problems in manifold valued BV -spaces

    work page

  14. [14]

    Babadjian, V

    J.-F. Babadjian, V. Millot , Homogenization of variational problems in manifold valued Sobolev Spaces, ESAIM: COCV 16 (2010), 833–855

    work page 2010

  15. [15]

    Ba\'ia, E

    M. Ba\'ia, E. Zappale , A note on the 3 D -2 D dimensional reduction of a micromagnetic thin film with nonhomogeneous profile , Appl. Anal. , 86 (5), (2007), 555--575

    work page 2007

  16. [16]

    and Fonseca I

    Baia M. and Fonseca I. , -convergence of functionals with periodic integrands via 2-scale convergence,

    work page

  17. [17]

    , Homogenization of variational problems in manifold valued Sobolev spaces, Control, Optimisation and Calculus of Variations, 16 (2010), 833–855

    Babadjian J-F., Milot V. , Homogenization of variational problems in manifold valued Sobolev spaces, Control, Optimisation and Calculus of Variations, 16 (2010), 833–855

    work page 2010

  18. [18]

    , A version of the fundamental theorem for Young measures, PDE's and continum models for phase transitions, Lecture notes physics, 334 (1989), 207-215

    Ball J.M. , A version of the fundamental theorem for Young measures, PDE's and continum models for phase transitions, Lecture notes physics, 334 (1989), 207-215

    work page 1989

  19. [19]

    : Lectures on Young measure theory and its applications in economics, Workshop on

    Balder, E.J. : Lectures on Young measure theory and its applications in economics, Workshop on

    work page

  20. [20]

    , Multiscale homogenization of convex functionals with discontinuous integrand, J

    Barchiesi M. , Multiscale homogenization of convex functionals with discontinuous integrand, J. Convex Anal. 14 (2007), 205-226

    work page 2007

  21. [21]

    Barchiesi , Loss of polyconvexity by homogenization: a new example, Calc

    M. Barchiesi , Loss of polyconvexity by homogenization: a new example, Calc. Var. and Partial Diff. Eq., 30 (2007), 215-230

    work page 2007

  22. [22]

    and Lions P

    Blanc X., Le Bris C. and Lions P. L. , Stochastic homogenization and random

    work page

  23. [23]

    Bethuel: approximation problem for Sobolev maps between two manifolds , Acta Math., 167 (1991), 153-206

    F. Bethuel: approximation problem for Sobolev maps between two manifolds , Acta Math., 167 (1991), 153-206

    work page 1991

  24. [24]

    Bethuel, X

    F. Bethuel, X. Zheng: of smooth functions between two manifolds in Sobolev spaces . Journal of functional analysis, 80(1) (1988), 60-75

    work page 1988

  25. [25]

    Bouchitt\'e, I

    G. Bouchitt\'e, I. Fonseca, L. Mascarenhas:

    work page

  26. [26]

    Braides, A

    A. Braides, A. Defranceschi , Homogenization of multiple integrals , Oxford Lecture Series in Mathematics and its Applications 12. Oxford University Press, New York, 1998

    work page 1998

  27. [27]

    Braides, A

    A. Braides, A. Defranceschi, A. Vitali:

    work page

  28. [28]

    Braides, I

    A. Braides, I. Fonseca, G. Francfort , 3D-2D Asymptotic Analysis for Inhomogeneous Thin Films, Indiana University Mathematics Journal, 49, (4) (2000), 1367-1404

    work page 2000

  29. [29]

    Buttazzo: Semicontinuity, relaxation, and integral representation in the calculus of variations

    G. Buttazzo: Semicontinuity, relaxation, and integral representation in the calculus of variations

    work page

  30. [30]

    Bukal, I

    M. Bukal, I. Vel ci\'c , On the simultaneous homogenization and dimension reduction in elasticity and locality of -closure, Calc. Var. Partial Differential Equations, 56 (3) (2017), 41 pp

    work page 2017

  31. [31]

    Carbone, K

    L. Carbone, K. Chacouche, A. Gaudiello:

    work page

  32. [32]

    Carita, A

    G. Carita, A. M. Ribeiro, E. Zappale , An Homogenization Result in W^ 1,p L^q . Journal of Convex Analysis, 18, No. 4, (2011), 1093-1126

    work page 2011

  33. [33]

    Chakrabortty, G

    A. Chakrabortty, G. Griso, J. Orlik , Dimension reduction and homogenization of composite plate with matrix pre-strain, Asymptot. Anal., 138 , (4), (2024), 255--310

    work page 2024

  34. [34]

    Chmara, J

    M. Chmara, J. Maksymiuk , Anisotropic Orlicz-Sobolev spaces of vector valued functions and Lagrange equations, J. Math. Anal. Appl. 456 , (2017), 457-475

    work page 2017

  35. [35]

    Cioranescu, A

    D. Cioranescu, A. Damlamian, G. Griso:

    work page

  36. [36]

    Cioranescu, A

    D. Cioranescu, A. Damlamian, G. Griso , The periodic unfolding method , Ser.\,Contemp.\,Math.\,3, (2018), 513 pp. Theory and Applications to Partial Differential Problems

    work page 2018

  37. [37]

    Dal Maso , An Introduction to -convergence

    G. Dal Maso , An Introduction to -convergence. Progress in Nonlinear Differential Equations and Their Applications Volume 8, 1993

    work page 1993

  38. [38]

    Dacorogna , Direct methods in the calculus of variations

    B. Dacorogna , Direct methods in the calculus of variations . Springer-Verlag, 2008

    work page 2008

  39. [39]

    Dacorogna, I

    B. Dacorogna, I. Fonseca, J. Mal\'y, K. Trivisa , Manifold constrained variational problems. Calc. Var. Part. Diff. Eq., 9 (1999), 185–206

    work page 1999

  40. [40]

    Eleuteri, L

    M. Eleuteri, L. Lussardi, A. Torricelli, E. Zappale , Homogenization and 3D-2D dimension reduction of a functional on manifold valued Sobolev spaces, Nonlinear Analysis: Real World Applications, 91 , 104579, (2026)

    work page 2026

  41. [41]

    Fonseca, S

    I. Fonseca, S. M\"uller:

    work page

  42. [42]

    Fabricius, M

    J. Fabricius, M. Gahn , Homogenization and dimension reduction of the Stokes problem with Navier-slip condition in thin perforated layers, Multiscale Model. Simul. 21 (4), (2023), 1502--1533

    work page 2023

  43. [43]

    Fatima, E

    T. Fatima, E. Ijioma, T. Ogawa, A. Muntean , Homogenization and dimension reduction of filtration combustion in heterogeneous thin layers, Netw. Heterog. Media 9 (4) (2014), 709-737

    work page 2014

  44. [44]

    Focardi , Semicontinuity of vectorial functionals in Orlicz-Sobolev spaces, Rend

    M. Focardi , Semicontinuity of vectorial functionals in Orlicz-Sobolev spaces, Rend. Istit. Math. Univ. Trieste,Vol. XXIX (1997) 141-161

    work page 1997

  45. [45]

    Fonseca, S

    I. Fonseca, S. M\"uller and P.\,Pedregal:

    work page

  46. [46]

    Fotso Tachago, G

    J. Fotso Tachago, G. Gargiulo, H. Nnang, E. Zappale , Multiscale Homogenization of integral convex functionals in Orlicz-Sobolev Setting Evolution Equations and Control Theory , 10 , n.2,(2021), 297 – 320

    work page 2021

  47. [47]

    Fotso Tachago, G

    J. Fotso Tachago, G. Gargiulo, H. Nnang, E. Zappale , Some convergence results on the periodic nfolding operator in Orlicz setting, Integral Methods in Science and Engineering, Conference Proceedings, (2023), 361-371

    work page 2023

  48. [48]

    Fotso Tachago, G

    J. Fotso Tachago, G. Gargiulo, H. Nnang, E. Zappale , Homogenization of non-convex integral energies with Orlicz growth, via periodic unfolding, J. Math. Anal. Appl. 552 , (2025), no. 1, Paper No. 129705, 22 pp

    work page 2025

  49. [49]

    Fotso Tachago, H

    J. Fotso Tachago, H. Nnang , Two scale convergence of integral functional with convex periodic and nonstandard growth integrands, Acta Appl. Math., 121, (2012), 175-196

    work page 2012

  50. [50]

    Fotso Tachago H

    J. Fotso Tachago H. Nnang, E. Zappale , Relaxation of Periodic and Nonstandard Growth Integrals by means of Two-scale convergence, Integral Methods in Science and Engineering., (2019), DOI: 10.1007/978-030-16077-7_ - 10

    work page doi:10.1007/978-030-16077-7_ 2019

  51. [51]

    Fotso Tachago H

    J. Fotso Tachago H. Nnang, E. Zappale , Reiterated periodic homogenization of integral functional with convex and nonstandard growth integrands, Opuscula Math., 41, (2021), 113-143

    work page 2021

  52. [52]

    & Nnang H

    Fotso Tachago J. & Nnang H. & Zappale E. , Reiterated homogenization of nonlinear elliptic operators with nonstandard growth,

    work page

  53. [53]

    Fotso Tachago H

    J. Fotso Tachago H. Nnang, E. Zappale , Reiterated Homogenization of nonlinear degenerate elliptic operators with nonstandard growth, Differential Integral Equations 37 (9/10), (2024), 717-752, DOI: 10.57262/die037-0910-717B

    work page doi:10.57262/die037-0910-717b 2024

  54. [54]

    Fotso Tachago, H

    J. Fotso Tachago, H. Nnang, E. Zappale , Reiterated periodic homogenization of integral functional with convex and nonstandard growth integrands, Opuscula Math., 41, (2021), 113-143

    work page 2021

  55. [55]

    Fotso Tachago Joel , Homogénéisation stochastique-périodique des équations de Maxwell, Th\` e se de Doctorat, Universit\' e de Yaound\' e 1, Cameroun, 2016-2017

    work page 2016

  56. [56]

    Fotso Tachago, H

    J. Fotso Tachago, H. Nnang, F. Tchinda, E. Zappale , (Two-scale) W^ 1 L^ -gradient Young measures and homogenization of integral functionals in Orlicz-Sobolev spaces , J Elliptic and Parabolic Equations, htpps://doi.org/10.1007/s41808-024-00294-4

    work page doi:10.1007/s41808-024-00294-4

  57. [57]

    Francfort, F

    G. Francfort, F. Murat, L. Tartar , Homogenization of monotone operators in divergence form with x-dependent multivalued graphs. Annali di Matematica Pura ed Applicata, 188, (2009), 631-652

    work page 2009

  58. [58]

    Gaudiello, R

    A. Gaudiello, R. Hadiji , Ferromagnetic thin multi-structures, J. Differential Equations , 257 (5), (2014), 1591--1622

    work page 2014

  59. [59]

    Gaudiello, R

    A. Gaudiello, R. Hadiji , Junction of ferromagnetic thin films Calc. Var. Partial Differential Equations 39 (3-4), (2010), 593–619

    work page 2010

  60. [60]

    Gioia, R

    G. Gioia, R. D. James , Micromagnetics of very thin films. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 453 (1956), (1997), 213-223

    work page 1956

  61. [61]

    Kristensen, F

    J. Kristensen, F. Rindler: affine approximations for functions

    work page

  62. [62]

    Kazarian , Estimates of maximal functions in the Orlicz spaces L_ ^ ( S ^ n-1 ) and L_ ^ ( R ^ n ) , Ivestiya Akademii Nauk Armenii Matematika, 28(3), 55-70, 1993

    S.S. Kazarian , Estimates of maximal functions in the Orlicz spaces L_ ^ ( S ^ n-1 ) and L_ ^ ( R ^ n ) , Ivestiya Akademii Nauk Armenii Matematika, 28(3), 55-70, 1993

    work page 1993

  63. [63]

    and Pedegral P

    Kinderlehrer D. and Pedegral P. , Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal., 4 (1988), 59-90

    work page 1988

  64. [64]

    and Pedegral P

    Kinderlehrer D. and Pedegral P. , Characterization of Young measures generated by gradients, Arch. Rational Mech. Anal., 115 (1991), 329-365

    work page 1991

  65. [65]

    Kozarzewski, E

    P.A. Kozarzewski, E. Zappale , Orlicz equi-integrability for scaled gradients, Journal of Elliptic and Parabolic Equations , 3 (2017), 1-2 , 1 – 13

    work page 2017

  66. [66]

    Kozarzewski, E

    P.A. Kozarzewski, E. Zappale , A note on Optimal design for thin structures in the Orlicz-Sobolev setting, Proceedings of the IMSE conference 2016, DOI 10.1007/978-3-319-59384-5-14

    work page doi:10.1007/978-3-319-59384-5-14 2016

  67. [67]

    Kreisbeck, S

    C. Kreisbeck, S. Kr\"omer , Heterogeneous thin films: combining homogenization and dimension reduction with directors, SIAM J. Math. Anal., 48 (2), (2016), 785-820

    work page 2016

  68. [68]

    Laskowski, H

    W. Laskowski, H. T. Nguyen , Effective energy integral functionals for thin films in the O rlicz- S obolev space setting , Demonstratio Math. , 46 , n. 3 (2013), 585--604

    work page 2013

  69. [69]

    Le Dret, A

    H. Le Dret, A. Raoult , The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. Journal de mathématiques pures et appliquées, 74 (6), (1995), 549-578

    work page 1995

  70. [70]

    Lussardi, A

    L. Lussardi, A. Torricelli, E. Zappale , Homogenization and 3D-2D dimension reduction of a functional on manifold valued BV space, to appear in Mathematics in Engineering

    work page

  71. [71]

    Mingione, D

    G. Mingione, D. Mucci , Integral functionals and the gap problem: sharp bounds for relaxation and energy concentration, SIAM J. Math. Anal. , 36, (2005), n. 5 1540--1579

    work page 2005

  72. [72]

    Neukamm , Homogenization, linearization and dimension reduction in elasticity with variational methods, Ph.D

    S.M. Neukamm , Homogenization, linearization and dimension reduction in elasticity with variational methods, Ph.D. Thesis, Technische Universit\"at M\"unchen, Zentrum Mathematik, 2010

    work page 2010

  73. [73]

    Nguetseng , A general convergence result for a functional related to the theory of homogenization

    G. Nguetseng , A general convergence result for a functional related to the theory of homogenization. SIAM Journal on Mathematical Analysis, 20 , (3), (1989), 608-623

    work page 1989

  74. [74]

    Nguetseng, H

    G. Nguetseng, H. Nnang, J.L. Woukeng, Deterministic homogenization of integral functionals with

    work page

  75. [75]

    Nguetseng, H

    G. Nguetseng, H. Nnang, J.L. Woukeng , Deterministic homogenization of integral functional with convex integrands, Nonlinear Differ. Equ. Appl. NoDEA, 17, (2010), 835-876

    work page 2010

  76. [76]

    Pisante , Homogenization of micromagnetics large bodies, ESAIM Control Optim

    G. Pisante , Homogenization of micromagnetics large bodies, ESAIM Control Optim. Calc. Var. , 10 . (2), (2004), 295--314

    work page 2004

  77. [77]

    Sanchez-Palencia , Fluid flow in porous media

    E. Sanchez-Palencia , Fluid flow in porous media. Non-homogeneous media and vibration theory, (1980), 129-157

    work page 1980

  78. [78]

    Séminaire Goulaouic-Schwartz

    L. Tartar , Homogénéisation et compacité par compensation. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi" Séminaire Goulaouic-Schwartz", (1977), 1-12

    work page 1977

  79. [79]

    Tartar , The general theory of homogenization: a personalized introduction

    L. Tartar , The general theory of homogenization: a personalized introduction. Springer Science & Business Media, 2009

    work page 2009