Existence of discontinuous absolute minima for certain multiple integrals without growth properties

@article{Cesari1988,
abstract = {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.},
author = {Cesari, Lamberto},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {BV function; Property (Q); Property (F); Serrin integral; multiple integrals; Serrin integrals},
language = {eng},
month = {12},
number = {4},
pages = {661-671},
publisher = {Accademia Nazionale dei Lincei},
title = {Existence of discontinuous absolute minima for certain multiple integrals without growth properties},
url = {http://eudml.org/doc/287250},
volume = {82},
year = {1988},
}

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. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1988/12//
PB - Accademia Nazionale dei Lincei
VL - 82
IS - 4
SP - 661
EP - 671
AB - 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.
LA - eng
KW - BV function; Property (Q); Property (F); Serrin integral; multiple integrals; Serrin integrals
UR - http://eudml.org/doc/287250
ER -