On the Dirichlet Problem with Orlicz Boundary Data

Zecca, Gabriella

Bollettino della Unione matematica italiana, Série 8, 10B (2007) no. 3, pp. 661-679

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{equation*} \begin{cases} Lu=0 \text{in } B\\ u_{|\partial B}=f \end{cases} \end{equation*}$B$ 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.