A general chain rule for distributional derivatives
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- by L. Ambrosio and G. Dal Maso PDF
- Proc. Amer. Math. Soc. 108 (1990), 691-702 Request permission
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
-
L. Ambrosio, A compactness theorem for a special class of functions of bounded variation (to appear in Boll. Un. Mat. Ital.).
- 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
- 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
- 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
- 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 151557
- C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310 G. Dal Maso, P. Le Floch, and F. Murat, (paper in preparation).
- 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 62214, DOI 10.1007/BF02412838
- 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 74499
- 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
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682, DOI 10.1007/978-1-4684-9486-0
- 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
- M. Marcus and V. J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal. 45 (1972), 294–320. MR 338765, DOI 10.1007/BF00251378
- Mario Miranda, Distribuzioni aventi derivate misure insiemi di perimetro localmente finito, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 27–56 (Italian). MR 165073
- Mario Miranda, Superfici cartesiane generalizzate ed insiemi di perimetro localmente finito sui prodotti cartesiani, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 18 (1964), 515–542 (Italian). MR 174706
- 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 192177 —, Équations elliptiques du second ordre à coefficients discontinus, Les Presses de l’Université de Montréal, Montréal, 1966. A. I. Vol’pert, The spaces BV and quasi-linear equations, Math. USSR-Sb. 2 (1967), 225-267.
- 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
Additional Information
- © Copyright 1990 American Mathematical Society
- 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