The parametric Weierstrass integral over a BV curve as a length functional

• Autorzy: Loris Faina
• Języki publikacji: EN

Abstrakty

EN

The constructive definition of the Weierstrass integral through only one limit process over finite sums is often preferable to the more sophisticated definition of the Serrin integral, especially for approximation purposes. By proving that the Weierstrass integral over a BV curve is a length functional with respect to a suitable metric, we discover a further natural reason for studying the Weierstrass integral. This characterization was conjectured by Menger.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1998
• Tom: 127
• Numer: 1
• Strony: 9-19

Daty

wydano

1998

otrzymano

1996-05-06

poprawiono

1997-06-17

Twórcy

autor

Loris Faina

• Dipartimento di Matematica, Università di Perugia, Via L. Vanvitelli, 1, 06123 Perugia, Italy,

Bibliografia

• [1] M. Boni, Variazione generalizzata con peso e quasi additività, Atti Sem. Mat. Fis. Univ. Modena 25 (1976), 195-210.
• [2] M. Boni and P. Brandi, Variazione, classica e generalizzata, con peso, ibid. 23 (1974), 1-22.
• [3] P. Brandi and A. Salvadori, Sull'integrale debole alla Burkill-Cesari, ibid. 27, (1978), 14-38.
• [4] P. Brandi and A. Salvadori, Martingale ed integrale alla Burkill-Cesari, Rend. Accad. Naz. Lincei 67 (1979), 197-203.
• [5] P. Brandi and A. Salvadori, Sull'area generalizzata, Atti Sem. Mat. Fis. Univ. Modena 24 (1979), 33-62.
• [6] P. Brandi and A. Salvadori, Un teorema di rappresentazione per l'integrale parametrico del Calcolo delle Variazioni alla Weierstrass, Ann. Mat. Pura Appl. 124 (1980), 39-58.
• [7] P. Brandi and A. Salvadori, Sull'estensione dell'integrale debole alla Burkill-Cesari ad una misura, Rend. Circ. Mat. Palermo 30 (1981), 207-234.
• [8] P. Brandi and A. Salvadori, Existence, semicontinuity and representation for the integrals of the Calculus of Variations. The BV case, in: Atti Convegno Celebrativo I Centenario Circolo Matematico di Palermo, 1984, 447-462.
• [9] P. Brandi and A. Salvadori, The nonparametric integral of the Calculus of Variations as a Weierstrass integral: Existence and representation, J. Math. Anal. Appl. 107 (1985), 67-95.
• [10] P. Brandi and A. Salvadori, 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), 319-325.
• [11] P. Brandi and A. Salvadori, On the lower semicontinuity of certain integrals of the Calculus of Variations, J. Math. Anal. Appl. 144 (1989), 183-205.
• [12] P. Brandi and A. Salvadori, A quasi-additivity type condition and the integral over a BV variety, Pacific J. Math. 146 (1990), 1-19.
• [13] P. Brandi and A. Salvadori, On the definition and properties of a variational integral over a BV curve, to appear.
• [14] J. C. Breckenridge, Burkill-Cesari integrals of quasi additive interval functions, Pacific J. Math. 37 (1971), 635-654.
• [15] L. Cesari, Sulle funzioni a variazione limitata, Ann. Scuola Norm. Sup. Pisa 5 (1936), 299-312.
• [16] L. Cesari, Quasi additive set functions and the concept of integral over a variety, Trans. Amer. Math. Soc. 102 (1962), 94-113.
• [17] L. Cesari, Extension problem for quasi additive set functions and Radon-Nikodym derivatives, ibid., 114-146.
• [18] K. Menger, Géométrie Générale, Mémorial Sci. Math. 124, Gauthier-Villars, Paris, 1954.
• [19] A. Salvadori, Sulla convergenza in lunghezza in senso generalizzato con peso per una successione di curve parametriche, Rend. Circ. Mat. Palermo 26 (1977), 195-228.
• [20] G. Warner, The generalized Weierstrass-type integral ∫f(ξ,ϕ), Ann. Scuola Norm. Sup. Pisa 22 (1968), 163-192.
• [21] G. Warner, The Burkill-Cesari integral, Duke Math. J. 35 (1968), 61-78.