zbMATH — the first resource for mathematics

A general chain rule for distributional derivatives. (English) Zbl 0685.49027
Summary: We prove a general chain rule for the distributional derivatives of the composite function \(v(x)=f(u(x))\), where u: \({\mathbb{R}}^ n\to {\mathbb{R}}^ m\) has bounded variation and f: \({\mathbb{R}}^ m\to {\mathbb{R}}^ k\) is Lipschitz continuous.

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
26B30 Absolutely continuous real functions of several variables, functions of bounded variation
46G05 Derivatives of functions in infinite-dimensional spaces
26B40 Representation and superposition of functions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI
[1] L. Ambrosio, A compactness theorem for a special class of functions of bounded variation (to appear in Boll. Un. Mat. Ital.). · Zbl 0767.49001
[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. · Zbl 0662.49007
[3] G. Anzellotti and M. Giaquinta, BV functions and traces, Rend. Sem. Mat. Univ. Padova 60 (1978), 1 – 21 (1979) (Italian, with English summary). · Zbl 0432.46031
[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). · Zbl 0524.35042
[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. · Zbl 0116.31804
[6] C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. · Zbl 0346.46038
[7] G. Dal Maso, P. Le Floch, and F. Murat, (paper in preparation).
[8] Ennio De Giorgi, Su una teoria generale della misura (\?-1)-dimensionale in uno spazio ad \? dimensioni, Ann. Mat. Pura Appl. (4) 36 (1954), 191 – 213 (Italian). · Zbl 0055.28504
[9] Ennio De Giorgi, Nuovi teoremi relativi alle misure (\?-1)-dimensionali in uno spazio ad \? dimensioni, Ricerche Mat. 4 (1955), 95 – 113 (Italian). · Zbl 0066.29903
[10] E. De Giorgi, F. Colombini, and L. C. Piccinini, Frontiere orientate di misura minima e questioni collegate, Scuola Normale Superiore, Pisa, 1972 (Italian). · Zbl 0296.49031
[11] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801
[12] Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. · Zbl 0545.49018
[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. · Zbl 0569.60053
[14] M. Marcus and V. J. Mizel, Absolute continuity on tracks and mappings of Sobolev spaces, Arch. Rational Mech. Anal. 45 (1972), 294 – 320. · Zbl 0236.46033
[15] Mario Miranda, Distribuzioni aventi derivate misure insiemi di perimetro localmente finito, Ann. Scuola Norm. Sup. Pisa (3) 18 (1964), 27 – 56 (Italian). · Zbl 0131.11802
[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). · Zbl 0152.24402
[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). · Zbl 0151.15401
[18] -, Équations elliptiques du second ordre à coefficients discontinus, Les Presses de l’Université de Montréal, Montréal, 1966. · Zbl 0151.15501
[19] A. I. Vol’pert, The spaces BV and quasi-linear equations, Math. USSR-Sb. 2 (1967), 225-267. · Zbl 0168.07402
[20] A. I. Vol\(^{\prime}\)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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.