Pith. sign in

REVIEW 5 minor 42 references

A matrix-valued measure associated to the derivatives of a function of generalised bounded deformation

T0 review · 0 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Every function of generalised bounded deformation admits a symmetric matrix-valued measure that replaces the distributional symmetric gradient, and the vanishing of its Cantor part exactly characterizes the space GSBD.

desk verdict Serious structural paper: the new matrix-valued measure and the GSBD characterization look right, with the real risk concentrated in the Section 6 grid estimates. read the letter →

arxiv 2506.19978 v1 pith:3MBWE2QS submitted 2025-06-24 math.FA math.AP

Gianni Dal Maso , Davide Donati This is my paper · ORCID
classification math.FAmath.AP MSC 49Q2074A45
keywords freediscontinuityproblemsfunctionsofgeneralisedboundeddeformationGSBDsymmetricgradientmatrix-valuedmeasureCantorpartslicingfineproperties
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that every function of generalised bounded deformation (GBD) carries a matrix-valued measure that does the work of a symmetric gradient even when no distributional gradient exists. For each threshold $r>0$ this measure $\mu_{u,r}$ is obtained as the common limit of the directional derivatives of suitably truncated projections $\tau_R(u\cdot\xi)$, and it splits into an absolutely continuous part with density the approximate symmetric gradient $Eu$, a jump part supported on the small jumps, and a Cantor part that is singular to Lebesgue measure and vanishes on sets that are $\sigma$-finite for $H^{d-1}$. The same construction characterizes the smaller space GSBD: a GBD function belongs to GSBD exactly when its Cantor part is zero. This matters for fracture mechanics, where GBD is the natural frame for cohesive models and no distributional gradient is available.

What carries the argument

The load-bearing object is the family of slice measures $$ \$\sigma$^\xi_u(B):=|\xi|\int_{\Pi_\xi} Du^\xi_y\big((B\setminus $J^{1}$_u)^\xi_y\big)\,$dH^{{d-1}}$(y), $$ which integrate the one-dimensional BV derivative of the slice $u(y+t\xi)\cdot\xi$, excluding points where the jump size is at least 1. The proof's central step is showing that for every Borel set $B$ the map $\xi\mapsto\sigma^\xi_u(B)$ is a quadratic form---2-homogeneous, lower bounded, and satisfying the parallelogram identity---so that by a standard characterization there is a symmetric matrix $\mu_u(B)$ with $\sigma^\xi_u(B)=\mu_u(B)\xi\cdot\xi$. In dimension 2 the parallelogram identity is proved by discretising the slice integrals on the grid $\{\omega+(i/k)\xi+(j/k)\eta\}$ and using a carefully chosen translation $\omega$ so that grid segments meeting the jump set contribute a vanishing error; dimensions $d>2$ follow by a Fubini-type reduction to two-dimensional slices.

What would settle it

Evaluate formula (1.5) for a GBD function whose singular part is supported on a purely unrectifiable set that is not aligned with the coordinate axes, using $\xi$, $\eta$, $\xi+\eta$, and $\xi-\eta$ in a Borel set $B$ meeting that set; if the four numbers violate the parallelogram identity, the claimed quadraticity---and hence the existence of $\mu_u$---fails.

Watch

Extended reading notes

Core claim

The central result is that the slice data of $u\in GBD(\Omega)$ can be assembled into one symmetric-matrix-valued bounded Radon measure. Theorem 8.1 and Corollary 8.6 give, for every $r>0$, a measure $\mu_{u,r}\in M_b(\Omega;\mathbb{R}^{d\times d}_{\mathrm{sym}})$ such that for every unit vector $\xi$ $$ \mu_{u,r}(B)\xi\cdot\xi=\lim_{R\to+\infty} D_\xi(\tau_R(u\cdot\xi))(B\setminus J^r_u) $$ for every Borel set $B$, with $\tau_R$ the piecewise-linear truncation at $\pm R/2$. The decomposition $$ \mu_{u,r}=\mu^a_u+\mu^c_u+\mu^j_{u,r} $$ holds with $\mu^a_u(B)=\int_B Eu\,dx$, $\mu^j_{u,r}(B)=\int_{(J_u\setminus J^r_u)\cap B}[u]\odot\nu_u\,dH^{d-1}$, and $\mu^c_u$ singular with respect to $L^d$ and zero on $H^{d-1}$-$\sigma$-finite Borel sets. Theorem 8.9 then characterises $GSBD(\Omega)$ as the subspace of $GBD(\Omega)$ consisting of functions with $\mu^c_u=0$, giving an intrinsic definition of GSBD that no longer refers to slicing.

Load-bearing premise

The argument hinges on a grid-shift estimate in the plane: for every scale one can shift the grid so that the total variation of the slices that meet the jump set is negligible; if this estimate fails for some admissible function, the parallelogram identity---and with it the existence of $\mu_u$---would collapse.

Editorial extensions

If this is right

  • For every $u\in GBD(\Omega)$ and $r>0$, the directional measures $\sigma^\xi_u(B)$ are the quadratic form of a single symmetric-matrix-valued measure $\mu_{u,r}$, so the limit in (1.3) exists simultaneously for all directions and all Borel sets avoiding $J^r_u$.
  • The three-part decomposition with explicit densities for the absolutely continuous and jump parts and a singular Cantor part vanishing on $H^{d-1}$-$\sigma$-finite sets holds for every $u\in GBD(\Omega)$, not just for $BD$.
  • $GSBD(\Omega)$ is exactly $\{u\in GBD(\Omega): \mu^c_u=0\}$, an intrinsic characterization independent of slicing.
  • Using a recent finite-direction characterization of GBD, an $L^d$-measurable function belongs to $GSBD(\Omega)$ as soon as the slice conditions hold for an orthonormal basis and their sums, with the quantitative bound $\lambda_u(\Omega)\leq C_d\Lambda$.
  • For $u\in BD(\Omega)$, $\mu_{u,r}$ equals the distributional symmetric gradient $Eu$ restricted to $\Omega\setminus J^r_u$, so the new measure extends the classical BD structure theorem.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if the decomposition is stable under the natural convergence for GBD, energies in cohesive fracture models could be written directly as sums over $\mu^a_u$, $\mu^c_u$, and $\mu^j_u$, making the Cantor part the term that governs diffuse damage.
  • Editorial inference: the same slice-integral-to-quadratic-form mechanism may give a general criterion for when other 2-homogeneous, slice-defined set functions are represented by matrix-valued measures, which would apply to relaxations of free-discontinuity energies.
  • Editorial inference: the finite-direction criterion (Theorem 8.10) suggests a concrete algorithmic test for GSBD membership: verify SBV slicing behavior in $d(d+1)/2$ directions and compute $\Lambda$; a positive test returns the quantitative bound $\lambda_u(\Omega)\leq C_d\Lambda$.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper associates to every function u in GBD(Ω) a symmetric-matrix-valued bounded Radon measure μ_u, and more generally a family μ_{u,r}, defined through the limits of the directional derivatives D_ξ(τ_R(u·ξ)) on sets avoiding the large-jump set J_u^r. The main result, Theorem 8.1 and Corollary 8.6, proves the existence of such measures and their three-part decomposition μ_{u,r} = μ_u^a + μ_u^c + μ_{u,r}^j, with explicit formulas for the absolutely continuous part ∫ Eu dx and the jump part ∫_{J_u\J_u^r} [u]⊙ν_u dH^{d-1}. In addition, Theorem 8.9 characterizes GSBD(Ω) as the set of u in GBD(Ω) for which μ_u^c = 0, giving an intrinsic, non-slicing characterization of GSBD. The proof is built on a lengthy quadraticity argument for the map ξ ↦ σ_u^ξ(B), proved first in dimension two by a discretization with careful control of bad grid indices, and then extended to higher dimensions by a Fubini-type slicing argument.

Significance. If correct, the paper closes a genuine gap in the theory of generalized bounded deformation: it provides a measure-theoretic analogue of the symmetric gradient Eu for functions that are not in BD(Ω), together with a Cantor part and a jump part. This is directly relevant to variational models of cohesive fracture, where GBD is the natural space once anti-plane symmetry is dropped. The characterization of GSBD by the vanishing of μ_u^c is a clean structural result that complements the slicing definition and the recent characterizations of Chambolle and Crismale. The proof is unusually detailed, with all technical lemmas and an appendix on measurability; the load-bearing estimate is Lemma 6.4, and I have checked its reduction to the uniform Riemann-sum bound (6.43) and the strip argument for h = 3,...,6. I find the argument coherent and do not see a surviving objection to the central claim. The paper is not machine-checked, so independent verification of the many algebraic identities in Sections 5–6 is advisable, but this is a matter of prudent verification, not a detected flaw.

minor comments (5)
  1. [Section 1] In the paragraph beginning 'The analysis of the fine properties...', the word 'reveales' should be 'reveals'.
  2. [Lemma 6.5, proof] In the proof of Lemma 6.5, the derivative is written as Du^ξ_y in two places ('Du^ξ_y({t}) = 0' and 'the measure µ := |Du^ξ_y| (U^ζ_y\J^ζ_y)'); for consistency with the statement of the lemma it should be Du^ζ_y.
  3. [Section 6, after (6.48)] The phrase 'ω = z1¯ζ + z2ζ with z1,∈ R' contains a stray comma and should read 'with z1 ∈ R'.
  4. [Section 6, Lemmas 6.4 and 6.8] The transliterations 'ˇCebyˇ s¨ ev' and 'Cebyˇ s¨ ev' should be normalized to a single spelling, for example 'Čebyšev' or 'Chebyshev'.
  5. [Theorem 8.1, Eq. (8.1)] The chain of equalities in (8.1) is correct because σ_u^ξ(J_u^1)=0 by definition (4.1), but the notation σ_u^ξ(B) = lim_{R→∞} D_ξ(τ_R(u·ξ))(B\J_u^1) is slightly compressed; a parenthetical reminder that the two occurrences of σ_u^ξ differ by a null set with respect to J_u^1 would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the measure is constructed from independently proved quadraticity of the slicing measures, and the GSBD characterization is a corollary of a derived slicing formula, not an input.

full rationale

The paper's central construction is not circular. The measure μ_u is defined after proving that ξ↦σ^ξ_u(B) is quadratic (Theorems 5.1 and 7.2), where σ^ξ_u is defined directly from the slicing measures Du^ξ_y (equation (4.1)). The proof of quadraticity rests on the Riemann-sum and grid estimates of Sections 5-6, especially Lemmas 6.4, 6.5, and 6.8; these estimates are internally coherent, are proved from the integrability of the slice jump-counting function (Lemma 2.4), and do not assume the existence of μ_u or the GSBD characterization. The decomposition μ_u = μ^a_u + μ^c_u + μ^j_u is then derived: the absolutely continuous part uses Theorem 2.12, the jump part uses Proposition 4.5, and the Cantor part uses Lemma A.2, all by explicit formulas rather than by fitting. The GSBD characterization in Theorem 8.9 is an equivalence obtained from the slicing formula (8.9), which expresses μ^c_u in terms of the one-dimensional Cantor parts D^c u^ξ_y; since μ^c_u is defined independently in Definition 8.3 and (8.9) is proved, this is a genuine characterization of the existing slicing definition of GSBD, not a self-definitional prediction. Self-citations to [20] supply the background definition and fine properties of GBD/GSBD, but those results do not contain the target conclusion and are used as standard external prerequisites. No fitted parameter is relabelled as a prediction, and no load-bearing claim reduces by construction to an earlier claim of the same paper.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants or ad hoc entities. Its axioms are the standard toolkit of geometric measure theory and the previously established fine properties and definition of GBD from [20, Theorem 2.12, Definition 2.5], the BD structure theory from [6], and the recent characterization from [13]. The only burden placed on the reader is accepting these cited results and the technical lemmas of Sections 3-7, which are proven in the paper.

assumptions (6)
  • domain assumption The fine properties of GBD functions stated in Theorem 2.12 (approximate symmetric gradient, rectifiability of Ju, slicing inclusions) from [20].
    Used throughout: in Definition 2.11, Proposition 4.3, Proposition 8.5, and elsewhere.
  • domain assumption Definition of GBD via slicing and existence of λ_u (Definition 2.5 and equation (2.20) from [20]).
    This is the starting point; the paper generalizes this framework.
  • domain assumption BD structure theorem from [6, Theorem 4.5] giving decomposition (1.2).
    Used as the benchmark in Remark 4.1 and Remark 8.2 to show the new measure agrees with Eu on BD.
  • domain assumption Decomposition of a GBD function into SBV plus GBD part (Proposition 4.6), citing [22].
    Reduces the proof of quadraticity to the case Ju≃J^1_u.
  • domain assumption Characterization of GBD from [13] (relied upon in Theorem 8.10).
    Used in the proof of Theorem 8.10 to conclude u∈GBD and the bound on λ_u.
  • standard math Standard BV slicing theory and geometric measure theory facts (e.g., Lemmas 2.4, 6.7, 7.1, and the Area Formula).
    Foundational tools used throughout the proof.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A matrix-valued measure associated to the derivatives of a function of generalised bounded deformation." pith.science (2026). https://pith.science/paper/3MBWE2QS

@misc{pith2026250619978,
  author       = {Pith},
  title        = {Pith review of: A matrix-valued measure associated to the derivatives of a function of generalised bounded deformation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3MBWE2QS}},
  note         = {Machine review of arXiv:2506.19978}
}
abstract

We associate to every function $u\in GBD(\Omega)$ a measure $\mu_u$ with values in the space of symmetric matrices, which generalises the distributional symmetric gradient $Eu$ defined for functions of bounded deformation. We show that this measure $\mu_u$ admits a decomposition as the sum of three mutually singular matrix-valued measures $\mu^a_u$, $\mu^c_u$, and $\mu^j_u$, the absolutely continuous part, the Cantor part, and the jump part, as in the case of $BD(\Omega)$ functions. We then characterise the space $GSBD(\Omega)$, originally defined only by slicing, as the space of functions $u\in GBD(\Omega)$ such that $\mu^c_u=0$.

Figures

Figures reproduced from arXiv: 2506.19978 by the authors.

Figure 1
Figure 1. The parallelogram U and the grid of points x k i,j associated to ω ∈ U and k = 3 Since the points x k i,j will be instrumental to the discretisation of the summands in (5.1), which are integrals over the straight lines Πζ for ζ ∈ {ξ, η, ξ + η, ξ − η}, we consider [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Projections of the points x k i,j onto the straight line Πξ+η It is clear from these definitions that for every ζ ∈ {ξ, η, ξ + η, ξ − η} and i, j ∈ Z there exists a unique real number t k,ζ i,j such that x k i,j = y k,ζ i,j + t k,ζ i,j ζ. (5.6) Let Cξ,η := |ξ| 2 |η| 2 − (ξ · η) 2 1/2 > 0. We observe that for i, j ∈ Z we have k|y k,ξ i,j+1 − y k,ξ i,j | = |π ξ (η)| = 1 |ξ| Cξ,η, k|y k,η i+1,j − y k,η i,j | = |π η (ξ… view at source ↗
Figure 3
Figure 3. The map x 7→ z k,ξ(x). By (6.11) and (6.12) for every y ∈ R 2 z k,ζ (y + tζ) = x k,ζ j (y) for every t ∈ [t k,ζ j (y), tk,ζ j+1(y)). (6.16) Geometrically (see [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 39 canonical work pages

  1. [1]

    S. Almi, E. Davoli, and M. Friedrich , Non-interpenetration conditions in the passage from non- linear to linearized Griffith fracture , J. Math. Pures Appl. (9), 175 (2023), pp. 1–36

    work page 2023

  2. [2]

    S. Almi, E. Davoli, A. Kubin, and E. Tasso, On De Giorgi’s conjecture of nonlocal approximations for free-discontinuity problems: The symmetric gradient case , 2024. arXiv https://arxiv.org/abs/ 2410.23908

    work page arXiv 2024

  3. [3]

    Almi and E

    S. Almi and E. Tasso, Brittle fracture in linearly elastic plates , Proc. Roy. Soc. Edinburgh Sect. A, 153 (2023), pp. 68–103

    work page 2023

  4. [4]

    , A new proof of compactness in G(S)BD , Adv. Calc. Var., 16 (2023), pp. 637–650

    work page 2023

  5. [5]

    Ambrosio, Existence theory for a new class of variational problems , Arch

    L. Ambrosio, Existence theory for a new class of variational problems , Arch. Rational Mech. Anal., 111 (1990), pp. 291–322

    work page 1990

  6. [6]

    Ambrosio, A

    L. Ambrosio, A. Coscia, and G. Dal Maso, Fine properties of functions with bounded deformation, Arch. Rational Mech. Anal., 139 (1997), pp. 201–238

    work page 1997

  7. [7]

    Ambrosio, N

    L. Ambrosio, N. Fusco, and D. Pallara , Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000

    2000

  8. [8]

    Bellettini, A

    G. Bellettini, A. Coscia, and G. Dal Maso , Compactness and lower semicontinuity properties in SBD(Ω), Math. Z., 228 (1998), pp. 337–351

    work page 1998

Show all 42 references
  1. [9]

    Bourdin, G

    B. Bourdin, G. A. Francfort, and J.-J. Marigo , The variational approach to fracture, Springer, New York, 2008. [reprinted from. J. Elasticity 91, 5–148 (2008)]

    2008

  2. [10]

    Caillet and F

    T. Caillet and F. Santambrogio , Doubly nonlinear diffusive PDEs: new existence results via generalized Wasserstein gradient flows , SIAM J. Math. Anal., 56 (2024), pp. 7043–7073

    2024

  3. [11]

    Chambolle, S

    A. Chambolle, S. Conti, and F. Iurlano , Approximation of functions with small jump sets and existence of strong minimizers of Griffith’s energy , J. Math. Pures Appl. (9), 128 (2019), pp. 119–139

    2019

  4. [12]

    Chambolle and V

    A. Chambolle and V. Crismale , Compactness and lower semicontinuity in GSBD, J. Eur. Math. Soc. (JEMS), 23 (2021), pp. 701–719

    2021

  5. [13]

    arXiv https://doi

    , A characterization of generalized functions of bounded deformation , 2025. arXiv https://doi. org/10.48550/arXiv.2502.10861

    2025 doi arXiv

  6. [14]

    , A general compactness theorem in G(S)BD, Indiana Univ. Math. J., 74 (2025), pp. 233–249

    2025

  7. [15]

    D. L. Cohn, Measure theory, Birkh¨ auser Advanced Texts: Basler Lehrb¨ ucher. [Birkh¨ auser Advanced Texts: Basel Textbooks], Birkh¨ auser/Springer, New York, second ed., 2013

    2013

  8. [16]

    Conti, M

    S. Conti, M. Focardi, and F. Iurlano , Existence of strong minimizers for the Griffith static fracture model in dimension two , Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 36 (2019), pp. 455– 474

    2019

  9. [17]

    Crismale and M

    V. Crismale and M. Friedrich , Equilibrium configurations for epitaxially strained films and mate- rial voids in three-dimensional linear elasticity, Arch. Ration. Mech. Anal., 237 (2020), pp. 1041–1098

    2020

  10. [18]

    Crismale, M

    V. Crismale, M. Friedrich, and J. Seutter, Adaptive finite element approximation for quasi-static crack growth, 2025. ArXiv: https://arxiv.org/abs/2503.18664

    2025 arXiv

  11. [19]

    Dal Maso, An introduction to Γ-convergence, vol

    G. Dal Maso, An introduction to Γ-convergence, vol. 8 of Progress in Nonlinear Differential Equations and their Applications, Birkh¨ auser Boston, Inc., Boston, MA, 1993

    1993

  12. [20]

    , Generalised functions of bounded deformation, J. Eur. Math. Soc. (JEMS), 15 (2013), pp. 1943– 1997

    2013

  13. [21]

    Dal Maso, G

    G. Dal Maso, G. A. Francfort, and R. Toader , Quasi-static evolution in brittle fracture: the case of bounded solutions , 2004. arXiv, https://arxiv.org/abs/math/0401198

    2004 arXiv

  14. [22]

    Dal Maso and R

    G. Dal Maso and R. Toader , Decomposition results for functions with bounded variation , Boll. Unione Mat. Ital. (9), 1 (2008), pp. 497–505

    2008

  15. [23]

    Paper No

    , A new space of generalised functions with bounded variation motivated by fracture mechanics , NoDEA Nonlinear Differential Equations Appl., 29 (2022). Paper No. 63, 36 pp

    2022

  16. [24]

    Convex Anal., 31 (2024), pp

    , Γ-convergence and integral representation for a class of free discontinuity functionals, J. Convex Anal., 31 (2024), pp. 411–476. 50 GIANNI DAL MASO AND DAVIDE DONATI

    2024

  17. [25]

    , Homogenisation problems for free discontinuity functionals with bounded cohesive surface terms, Arch. Ration. Mech. Anal., 248 (2024). Paper No. 109, 48 pp

    2024

  18. [26]

    Del Nin, Rectifiability of the jump set of locally integrable functions , Ann

    G. Del Nin, Rectifiability of the jump set of locally integrable functions , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 22 (2021), pp. 1233–1240

    2021

  19. [27]

    J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953

    1953

  20. [28]

    Federer , Geometric measure theory , vol

    H. Federer , Geometric measure theory , vol. Band 153 of Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag New York, Inc., New York, 1969

    1969

  21. [29]

    W. M. Feldman and K. Stinson , Compactness for GSBV p via concentration-compactness, 2025. arXiv, https://arxiv.org/abs/2501.16308

    2025 arXiv

  22. [30]

    G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem , J. Mech. Phys. Solids, 46 (1998), pp. 1319–1342

    1998

  23. [31]

    Friedrich, A compactness result in GSBV p and applications to Γ-convergence for free disconti- nuity problems, Calc

    M. Friedrich, A compactness result in GSBV p and applications to Γ-convergence for free disconti- nuity problems, Calc. Var. Partial Differential Equations, 58 (2019). Paper No. 86, 31 pp

    2019

  24. [32]

    Friedrich, C

    M. Friedrich, C. Labourie, and K. Stinson , Strong existence for free discontinuity problems in linear elasticity, SIAM J. Math. Anal., 57 (2025), pp. 1652–1679

    2025

  25. [33]

    Friedrich and F

    M. Friedrich and F. Solombrino , Quasistatic crack growth in 2d-linearized elasticity , Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, 35 (2018), pp. 27–64

    2018

  26. [34]

    Hahn, ¨Uber Ann¨ aherung an Lebesgue’sche Integrale durch Riemann’sche Summen, Sitzungsber

    H. Hahn, ¨Uber Ann¨ aherung an Lebesgue’sche Integrale durch Riemann’sche Summen, Sitzungsber. Math. Phys. Kl. K. Akad. Wiss. Wien, 123 (1914), pp. 713–743

    1914

  27. [35]

    Haj lasz, On approximate differentiability of functions with bounded deformation , Manuscripta Math., 91 (1996), pp

    P. Haj lasz, On approximate differentiability of functions with bounded deformation , Manuscripta Math., 91 (1996), pp. 61–72

    1996

  28. [36]

    Simon, Lectures on geometric measure theory, vol

    L. Simon, Lectures on geometric measure theory, vol. 3 of Proceedings of the Centre for Mathematical Analysis, Australian National University, Australian National University, Centre for Mathematical Analysis, Canberra, 1983

    1983

  29. [37]

    Suquet, Existence et r´ egularit´ e des solutions des ´ equations de la plasticit´ e, C

    P.-M. Suquet, Existence et r´ egularit´ e des solutions des ´ equations de la plasticit´ e, C. R. Acad. Sci. Paris S´ er. A-B, 286 (1978), pp. A1201–A1204

    1978

  30. [38]

    Tasso, On the continuity of the trace operator in GSBV (Ω) and GSBD(Ω), ESAIM Control Optim

    E. Tasso, On the continuity of the trace operator in GSBV (Ω) and GSBD(Ω), ESAIM Control Optim. Calc. Var., 26 (2020). Paper No. 30, 34 pp

    2020

  31. [39]

    , Weak formulation of elastodynamics in domains with growing cracks , Ann. Mat. Pura Appl. (4), 199 (2020), pp. 1571–1595

    2020

  32. [40]

    Temam , Probl` emes math´ ematiques en plasticit´ e, vol

    R. Temam , Probl` emes math´ ematiques en plasticit´ e, vol. 12 of M´ ethodes Math´ ematiques de l’Informatique [Mathematical Methods of Information Science], Gauthier-Villars, Montrouge, 1983

    1983

  33. [41]

    Temam and G

    R. Temam and G. Strang , Functions of bounded deformation , Arch. Rational Mech. Anal., 75 (1980), pp. 7–21

    1980

  34. [42]

    Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge Univer- sity Press, Cambridge, 1991

    D. Williams, Probability with martingales, Cambridge Mathematical Textbooks, Cambridge Univer- sity Press, Cambridge, 1991. SISSA, via Bonomea 265, Trieste, Italy Email address: dalmaso@sissa.it SISSA, via Bonomea 265, Trieste, Italy Email address: ddonati@sissa.it

    1991

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.