Discontinuous solutions in problems of optimization

Cesari, L.; Brandi, P.; Salvadori, A.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 15 (1988) no. 2, pp. 219-237.

[1] G. Anzellotti - M. Giaquinta, Funzioni BV e tracce, Rend. Sem. Mat. Padova 60 (1978), pp. 1-21. | Numdam | MR | Zbl

[2] P. Brandi - A. Salvadori, (a) "Sull'integrale debole alla Burkill-Cesari", Atti Sem. Mat. Fis. Univ. Modena 27 (1978), pp. 14-38; (b) "Existence, semicontinuity and representation for the integrals of the Calculus of Variations. The BV case", Atti Convegno celebrativo I centenario Circolo Matematico di Palermo (1984), pp. 447-462; (c) "L'integrale del Calcolo delle Variazioni alla Weierstrass lungo curve BV e confronto con i funzionali integrali di Lebesgue e Serrin", Atti Sem. Mat. Fis. Univ. Modena 35 (1987); (d) "A quasi-additivity type condition and the integral over a BV variety", Pacific Math. Journ., to appear; (e) "On the non-parametric integral over a BV surface", Nonlinear Analysis, to appear; (f) "On the lower semicontinuity of certain integrals of the Calculus of Variations", Journ. Math. Anal. Appl., to appear; (g) "On Weierstrass-type variational integrals over BV varieties", Rend. Accad. Naz. Lincei Roma, to appear; (h) "On the definition and properties of a variational integral over a BV curve", to appear. | MR

[3] J.C. Breckenridge, "Burkill-Cesari integrals of quasi additive interval functions", Pacific J. Math. 37 (1971), pp. 635-654. | MR | Zbl

[4] F. Cafiero, Criteri di compattezza per le successioni di funzioni generalmente a variazione limitata, Atti Accad. Naz. Lincei 8 (1950), pp. 305-310. | MR | Zbl

[5] L. Cesari, Sulle funzioni a variazione limitata, Annali Scuola Norm. Sup. Pisa (2)5, 1936, pp. 299-313. | JFM | Numdam | Zbl

[6] L. Cesari, (a) Sulle funzioni di due variabili a variazione limitata e sulla convergenza delle serie doppie di Fourier, Rend. Sem. Mat. Univ. Roma 1, 1937, pp. 277-294; (b) Sulle funzioni di più variabili a variazione limitata e sulla convergenza delle relative serie multiple di Fourier, Pontificia Accad. Scienze, Commentationes 3, 1939, pp. 171-197. | JFM | MR

[7] L. Cesari, Optimization - Theory and Applications. Problems with Ordinary Differential Equations, Springer Verlag 1983. | MR | Zbl

[8] L. Cesari, (a) "Quasi additive set functions and the concept of integral over a variety", Trans. Amer. Math. Soc. 102 (1962), pp. 94-113; (b) "Extension problem for quasi additive set functions and Radon-Nikodym derivatives", Trans. Amer. Math. Soc. 102 (1962), pp. 114-146. | MR

[9] L. Cesari, (a) Existence of discontinuous absolute minima for certain multiple integrals without growth properties, Rend. Accad. Naz. Lincei, to appear; (b) Existence of BV discontinuous absolute minima for modified multiple integrals of the calculus of variations, Rend. Circolo Mat. Palermo, to appear; (c) Rankine-Hugoniot type properties in terms of the calculus of variations for BV solutions, to appear. | MR

[10] L. Cesari - P. Brandi - A. Salvadori, (a) Existence theorems concerning simple integrals of the calculus of variations for discontinuous solutions, Archive Rat. Mech. Anal. 98, 1987, pp. 307-328; (b) Existence theorems for multiple integrals of the calculus of variations for discontinuous solutions, Annali Mat. Pura Appl. 153 (1989); (c) Seminormality conditions in the calculus of variations for BV solutions, to appear. | MR | Zbl

[11] L. Cesari - P. Pucci, (a) Remarks on discontinuous optimal solutions for simple integrals of the calculus of variations, Atti Sem. Mat. Fis. Univ. Modena, to appear; (b) Existence of BV discontinuous solutions of the Cauchy problem for conservation laws, to appear. | MR

[12] E. Conway - J. Smoller, Global solutions of the Cauchy problem for quasi linear first order equations in several space variables, Comm. Pure Appl. Math. 19, 1966, pp. 95-105. | MR | Zbl

[13] C.M. Dafermos, Generalized characteristics and the structure of solutions of hyperbolic conservation laws, Indiana Univ. Math. J. 26, 1977, pp. 1097-1119. | MR | Zbl

[14] R.J. Di Perna, Singularities of solutions of nonlinear hyperbolic systems of conservation laws, Archive Rat. Mech. Anal. 60, 1974, pp. 75-100. | MR | Zbl

[15] W.H. Fleming, Functions with generalized gradient and generalized surfaces, Annali Mat. Pura Appl. 44, 1957, pp. 93-103. | MR | Zbl

[16] E. Gagliardo, Proprietà di alcune classi di funzioni di più variabili, Ricerche Mat. 7, 1959, pp. 24-51. | MR | Zbl

[17] K. Krickeberg, Distributionen, Funktionen beschränkter Variation und Lebesguescher Inhalt nichtparametrischer Fldchen, Annali Mat. Pura Appl. 44, 1957, pp. 105-133. | MR | Zbl

[18] O.A. Oleinik, On discontinuous solutions of nonlinear differential equations, Uspekhi Mat. Nauk 12, 1957, pp. 3-73. English translation, Amer. Math. Soc. Transl. (2) 26, pp. 95-172. | MR | Zbl

[19] A. Salvadori, Theorems of lower semicontinuity, to appear.

[20] J. Serrin, (a) On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc. 101, 1961, pp. 139-167; (b) On the differentiability of functions of several variables, Archive Rat. Mech. Anal. 7, 1961, pp. 359-372. | MR

[21] C. Vinti, Non linear integration and Weierstrass integral over a manifold. Connection with theorems on martingales, Journ. Optimization Theory Appl. 41, 1983, pp. 213-237. | MR | Zbl

[22] A.L. Volpert, The space BV and quasi linear equations, Mat. Sb. 73, 1967, pp. 225-267.

[23] G. Warner, (a) The Burkill-Cesari integral, Duke Math. J. 35, 1968, pp. 61-78; (b) The generalized Weierstrass-type integral ∫ f (ξ, Φ), Annali Scuola Norm. Sup. Pisa 22, 1968, pp. 163-192. | Numdam