Existence and boundedness of minimizers of a class of integral functionals

Mercaldo, A.

Bollettino della Unione matematica italiana, Série 8, 6B (2003) no. 1, pp. 125-139.

Abstract (VO)

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 $0\le \alpha$, $1\le {\beta }_{1}p$, $0\le {\beta }_{2}p$, $\alpha +{\beta }_i\le p$, $a(x)$, $b_i(x)$, $g_i(x)$ ($i=1$, $2$) 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 $u_0\in W^{1,p}(a)\cap L^{\mathrm{\infty }}(\mathrm{\Omega })$.

Riassunto

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 $0\le \alpha$, $1\le {\beta }_{1}p$, $0\le {\beta }_{2}p$, $\alpha +{\beta }_i\le p$, $a(x)$, $b_i(x)$, $g_i(x)$ ($i=1$, $2$) 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 $u_0\in W^{1,p}(a)\cap L^{\mathrm{\infty }}(\mathrm{\Omega })$.

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