Existence and boundedness of minimizers of a class of integral functionals

Mercaldo, A.

Mercaldo, A.

Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003), p. 125-139 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

In this paper we consider a class of integral functionals whose integrand satisfies growth conditions of the type \begin{gather*} f(x, \eta, \xi) \geq a(x) \frac{|\xi|^{p}}{(1 + |\eta|)^{\alpha}} - b_{1}(x)|\eta|^{\beta_{1}}-g_{1}(x),\\ f(x, \eta, 0)\leq b_{2}(x)|\eta|^{\beta_{2}}+ g_{2}(x), \end{gather*} where $0 \leq α < p$ , $1 \leq β_{1} < p$ , $0 \leq β_{2} < p$ , $α + β_{i} \leq p$ , $a (x)$ , $b_{i} (x)$ , $g_{i} (x)$ ( $i = 1$ , $2$ ) are nonnegative functions satisfying suitable summability assumptions. We prove the existence and boundedness of minimizers of such a functional in the class of functions belonging to the weighted Sobolev space $W^{1, p} (a)$ , which assume a boundary datum $u_{0} \in W^{1, p} (a) \cap L^{\infty} (Ω)$ .

In questo lavoro si considera una classe di funzionali integrali, il cui integrando verifica le seguenti condizioni \begin{gather*} f(x, \eta, \xi) \geq a(x) \frac{|\xi|^{p}}{(1 + |\eta|)^{\alpha}} - b_{1}(x)|\eta|^{\beta_{1}}-g_{1}(x),\\ f(x, \eta, 0)\leq b_{2}(x)|\eta|^{\beta_{2}}+ g_{2}(x), \end{gather*} dove $0 \leq α < p$ , $1 \leq β_{1} < p$ , $0 \leq β_{2} < p$ , $α + β_{i} \leq p$ , $a (x)$ , $b_{i} (x)$ , $g_{i} (x)$ ( $i = 1$ , $2$ ) sono funzioni non negative che soddisfano opportune ipotesi di sommabilità. Si dimostra l'esistenza e la limitatezza di minimi di tali funzionali nella classe di funzioni appartenenti allo spazio di Sobolev pesato $W^{1, p} (a)$ , che assumono un assegnato dato al bordo $u_{0} \in W^{1, p} (a) \cap L^{\infty} (Ω)$ .

Publié le : 2003-02-01

MR 1955700 | Zbl 1150.49001

@article{BUMI_2003_8_6B_1_125_0,
     author = {A. Mercaldo},
     title = {Existence and boundedness of minimizers of a class of integral functionals},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {6-A},
     year = {2003},
     pages = {125-139},
     zbl = {1150.49001},
     mrnumber = {1955700},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2003_8_6B_1_125_0}
}

Mercaldo, A. Existence and boundedness of minimizers of a class of integral functionals. Bollettino dell'Unione Matematica Italiana, Tome 6-A (2003) pp. 125-139. http://gdmltest.u-ga.fr/item/BUMI_2003_8_6B_1_125_0/

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