On the Dirichlet Problem with Orlicz Boundary Data

Zecca, Gabriella

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

Abstract (VO)

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.

Riassunto

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{equation*}\begin{cases} Lu=0 \text{in } B\\ u_{|\partial B}=f \end{cases} \end{equation*} sia $L^\Phi$-risolubile. La risolubilità per $f \in L^\Phi$ intesa nel senso di [5], [8], dove viene trattato il caso $\Phi(t) = t^p$.

Bibliographie

[1] D. W. Boyd, Indices of function spaces and their relationship to interpolation, Canad. J. Math., 21 (1969), 1245-1254. | DOI | MR | Zbl

[2] S. Buckley, Estimates for operator norms on weighted space and reverse Jensen inequalities, Trans. Amer. Math. Soc., 340, 1 (1993), 253-257. | DOI | MR | Zbl

[3] R. R. Coifman - C. Fefferman, Weighted norm inqualities for the maximal functions and singular integrals, Studia Math., 54 (1974), 221-237. | DOI | MR

[4] A. P. Calderón, Inqualities for the maximal functions relative to a metric, Studia Math., 49 (1976), 297-306. | fulltext EuDML | DOI | MR

[5] B. E. J. Dahlberg, On estimates of harmonic measure, Arch. Rat. Mech. Anal., 65 (1977), 272-288. | DOI | MR | Zbl

[6] B. E. J. Dahlberg, On the Poisson integral for Lipschitz and $C^1$ domains, Studia Math., 66 (1979), 7-24. | fulltext EuDML | DOI | MR | Zbl

[7] J. Garcia-Cuerva - J. L. Rubio De Francia, Weighted norm inqualities and related topics, North-Holland Math. Stud., vol. 116, North-Holland, Amsterdam, (1985). | MR | Zbl

[8] Carlos E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, Conference Board of the Mathematical Sciences, Amer. Math. Soc. 83 (1991). | DOI | MR

[9] R. A. Kerman - A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math., 71 (1982), 277-284. | fulltext EuDML | DOI | MR | Zbl

[10] W. Matuszewska - W. Orlicz, On certain properties of W-functions, Bull. Acad. Polon. Sci., 8, 7 , (1960), 439-443. | MR | Zbl

[11] G. Moscariello - C. Sbordone, $a_\infty$ as a limit case of reverse - Hölder inequality when the exponent tends to 1, Ricerche Mat., XLIV, 1 (1995), 131-144. | MR | Zbl

[12] B. Muckenhoupt, Weighted norm inqualities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226. | DOI | MR | Zbl

[13] E. M. Stein - G. Weiss, An extension of a theorem of Marcinkiewicz and some of its applications, J. Math. Mech., 8 (1959), 263-264. | MR | Zbl

[14] G. Zecca, The unsolvability of the Dirichlet problem with $L(\log L)^a$ boundary data, Rend. Acc. Sc. Fis. Mat. Napoli, 72 (2005), 71-80. | MR | Zbl