On the Dirichlet Problem with Orlicz Boundary Data

Zecca, Gabriella

Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007), p. 661-679 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

Let us consider a Young's function $\Phi\colon\mathbb{R}^{+}\to\mathbb{R}^{+}$ satisfying the $\Delta_{2}$ condition together with its complementary function $\Psi$ , and let us consider the Dirichlet problem for a second order elliptic operator in divergence form: $\begin{cases}Lu=0&\text{in }B\\ u_{|\partial B}=f\end{cases}$ $B$ the unit ball of $\mathbb{R}^{n}$ . In this paper we give a necessary and sufficient condition for the $L^{\phi}$ -solvability of the problem, where $L^{\phi}$ is the Orlicz Space generated by the function $\Phi$ . This means solvability for $f\in L^{\Phi}$ in the sense of [5], [8], where the case $\Phi(t)=t^{p}$ is treated.

Sia $\Phi\colon\mathbb{R}^{+}\to\mathbb{R}^{+}$ una funzione di Young che soddisfa, con la sua funzione complementare $\Psi$ , la condizione $\Delta_{2}$ e siano $L^{\Phi}$ lo spazio di Orlicz generato dalla funzione $\Phi$ e $B$ la palla unitaria di $\mathbb{R}^{n}$ . Si presenta una condizione necessaria e sufficiente affinché il problema di Dirichlet per un operatore del secondo ordine ellittico in forma di divergenza: $\begin{cases}Lu=0&\text{in }B\\ u_{|\partial B}=f\end{cases}$ sia $L^{\Phi}$ -risolubile. La risolubilità per $f\in L^{\Phi}$ intesa nel senso di [5], [8], dove viene trattato il caso $\Phi(t)=t^{p}$ .

Publié le : 2007-10-01

MR 2351536 | Zbl 1177.35060

@article{BUMI_2007_8_10B_3_661_0,
     author = {Gabriella Zecca},
     title = {On the Dirichlet Problem with Orlicz Boundary Data},
     journal = {Bollettino dell'Unione Matematica Italiana},
     volume = {10-A},
     year = {2007},
     pages = {661-679},
     zbl = {1177.35060},
     mrnumber = {2351536},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BUMI_2007_8_10B_3_661_0}
}

Zecca, Gabriella. On the Dirichlet Problem with Orlicz Boundary Data. Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007) pp. 661-679. http://gdmltest.u-ga.fr/item/BUMI_2007_8_10B_3_661_0/

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