Geodesic
Existence of discontinuous absolute minima for certain multiple integrals without growth properties
Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 4, pp. 661-671

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

Zbl   MR

In the present paper the author discusses certain multiple integrals $I(u)$ of the calculus of variations satisfying convexity conditions, and no growth property, and the corresponding Serrin integrals $\mathfrak{I}(u)$, to which the existence theorems in [3,4,5] do not apply. However, in the present paper, the integrals $I(u)$ and $\mathfrak{I}(u)$ are reduced to simpler form $H(v)$ and $\mathcal{H}(v)$ to which the existence theorems above apply. Thus, we derive that $I(u) \le \mathfrak{I}(u)$, $H(v) \le \mathcal{H}(v)$, we obtain the existence of the absolute minimum for the Serrin forms $\mathfrak{I}(u)$ and $\mathcal{H}(v)$, and such minimum is given by BV functions, possibly discontinuous and not of Sobolev. 
Cesari, Lamberto. Existence of discontinuous absolute minima for certain multiple integrals without growth properties. Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 82 (1988) no. 4, pp. 661-671. http://geodesic.mathdoc.fr/item/RLINA_1988_8_82_4_a5/
@article{RLINA_1988_8_82_4_a5,
     author = {Cesari, Lamberto},
     title = {Existence of discontinuous absolute minima for certain multiple integrals without growth properties},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali},
     pages = {661--671},
     year = {1988},
     volume = {Ser. 8, 82},
     number = {4},
     zbl = {0736.49029},
     mrnumber = {1139813},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLINA_1988_8_82_4_a5/}
}
TY  - JOUR
AU  - Cesari, Lamberto
TI  - Existence of discontinuous absolute minima for certain multiple integrals without growth properties
JO  - Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali
PY  - 1988
SP  - 661
EP  - 671
VL  - 82
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/RLINA_1988_8_82_4_a5/
LA  - en
ID  - RLINA_1988_8_82_4_a5
ER  - 
%0 Journal Article
%A Cesari, Lamberto
%T Existence of discontinuous absolute minima for certain multiple integrals without growth properties
%J Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali
%D 1988
%P 661-671
%V 82
%N 4
%U http://geodesic.mathdoc.fr/item/RLINA_1988_8_82_4_a5/
%G en
%F RLINA_1988_8_82_4_a5

[1] Cesari L. (1936) - Sulle funzioni a variazione limitata, «Annali Scuola Norm. Sup. Pisa» (2) 5, 299-313. | fulltext EuDML | fulltext mini-dml | Jbk 62.0247.03 | MR

[2] Cesari L. (1983) - Optimization-Theory and Applications, Springer Verlag. | DOI | MR | Zbl

[3] Cesari L., Brandi P. e Salvatori A. (1988) - Discontinuous solutions in problems of optimization, «Annali Scuola Norm. Sup. Pisa», 15, 219-237. | fulltext EuDML | fulltext mini-dml | MR | Zbl

[4] Cesari L., Brandi P. e Salvadori A. (1987) - Existence theorems concerning simple integrals of the calculus of variations for discontinuous solutions, «Archive Rat. Mech. Anal.», 98, 307-328. | DOI | MR | Zbl

[5] Cesari L., Brandi P. e Salvadori A. (1988) - Existence theorems for multiple integrals of the calculus of variations for discontinuous solutions, «Annali Mat. Pura Appl.», (4), 152, 95-121. | DOI | MR | Zbl

[6] Cesari L., Brandi P. e Salvadori A., Seminormality conditions in the calculus of variations for BV solutions, to appear. | DOI | MR | Zbl

[7] Cesari L. e Pucci P. (1989) - Remarks on discontinuous optimal solutions for simple integrals of the calculus of variations, «Atti Sem. Mat. Fis. Univ. Modena», 37, 335-379. | MR | Zbl

[8] Krickeberg K. (1957) - Distributions, Funktionen beschränkter Variation und Lehesguescher Inhalt nichtparametrischer Flächen, — «Annali Mat. Pura Appl.», (4) 44, 92, 105-133. | MR | Zbl