Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A general chain rule for distributional derivatives


Authors: L. Ambrosio and G. Dal Maso
Journal: Proc. Amer. Math. Soc. 108 (1990), 691-702
MSC: Primary 26B30; Secondary 46F10, 49F22
DOI: https://doi.org/10.1090/S0002-9939-1990-0969514-3
MathSciNet review: 969514
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a general chain rule for the distribution derivatives of the composite function $ \upsilon (x) = f(u(x))$, where $ u:{{\mathbf{R}}^n} \to {{\mathbf{R}}^m}$ has bounded variation and $ f:{{\mathbf{R}}^m} \to {{\mathbf{R}}^k}$ is Lipschitz continuous.


References [Enhancements On Off] (What's this?)

  • [1] L. Ambrosio, A compactness theorem for a special class of functions of bounded variation (to appear in Boll. Un. Mat. Ital.).
  • [2] L. Ambrosio, S. Mortola, and V. M. Tortorelli, Functionals with linear growth defined on vector valued BV functions, J. Math. Pures Appl. (9) 70 (1991), no. 3, 269–323. MR 1113814
  • [3] G. Anzellotti and M. Giaquinta, BV functions and traces, Rend. Sem. Mat. Univ. Padova 60 (1978), 1–21 (1979) (Italian, with English summary). MR 555952
  • [4] Lucio Boccardo and François Murat, Remarques sur l’homogénéisation de certains problèmes quasi-linéaires, Portugal. Math. 41 (1982), no. 1-4, 535–562 (1984) (French, with English summary). MR 766874
  • [5] A.-P. Calderón and A. Zygmund, On the differentiability of functions which are of bounded variation in Tonelli’s sense, Rev. Un. Mat. Argentina 20 (1962), 102–121. MR 0151557
  • [6] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
  • [7] G. Dal Maso, P. Le Floch, and F. Murat, (paper in preparation).
  • [8] Ennio De Giorgi, Su una teoria generale della misura (𝑟-1)-dimensionale in uno spazio ad 𝑟 dimensioni, Ann. Mat. Pura Appl. (4) 36 (1954), 191–213 (Italian). MR 0062214, https://doi.org/10.1007/BF02412838
  • [9] Ennio De Giorgi, Nuovi teoremi relativi alle misure (𝑟-1)-dimensionali in uno spazio ad 𝑟 dimensioni, Ricerche Mat. 4 (1955), 95–113 (Italian). MR 0074499
  • [10] E. De Giorgi, F. Colombini, and L. C. Piccinini, Frontiere orientate di misura minima e questioni collegate, Scuola Normale Superiore, Pisa, 1972 (Italian). MR 0493669
  • [11] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • [12] Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682
  • [13] G. Letta, Martingales et intégration stochastique, Scuola Normale Superiore di Pisa. Quaderni. [Publications of the Scuola Normale Superiore of Pisa], Scuola Normale Superiore, Pisa, 1984 (French). With an appendix by F. Fagnola. MR 804390
  • [14] M. Marcus and V. J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal. 45 (1972), 294–320. MR 0338765, https://doi.org/10.1007/BF00251378
  • [15] Mario Miranda, Distribuzioni aventi derivate misure insiemi di perimetro localmente finito, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 27–56 (Italian). MR 0165073
  • [16] Mario Miranda, Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 515–542 (Italian). MR 0174706
  • [17] Guido Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 189–258 (French). MR 0192177
  • [18] -, Équations elliptiques du second ordre à coefficients discontinus, Les Presses de l'Université de Montréal, Montréal, 1966.
  • [19] A. I. Vol'pert, The spaces BV and quasi-linear equations, Math. USSR-Sb. 2 (1967), 225-267.
  • [20] A. I. Vol′pert and S. I. Hudjaev, Analysis in classes of discontinuous functions and equations of mathematical physics, Mechanics: Analysis, vol. 8, Martinus Nijhoff Publishers, Dordrecht, 1985. MR 785938

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26B30, 46F10, 49F22

Retrieve articles in all journals with MSC: 26B30, 46F10, 49F22


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-0969514-3
Article copyright: © Copyright 1990 American Mathematical Society