On the Dirichlet Problem with Orlicz Boundary Data

Gabriella Zecca

Bollettino dell'Unione Matematica Italiana (2007)

  • Volume: 10-B, Issue: 3, page 661-679
  • ISSN: 0392-4041

Abstract

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Let us consider a Young's function Φ : ℝ + → ℝ + satisfying the Δ 2 condition together with its complementary function Ψ , and let us consider the Dirichlet problem for a second order elliptic operator in divergence form: { L ⁢ u = 0 in  ⁢ B u | ∂ B = f B the unit ball of ℝ n . In this paper we give a necessary and sufficient condition for the L ϕ -solvability of the problem, where L ϕ is the Orlicz Space generated by the function Φ . This means solvability for f ∈ L Φ in the sense of [5], [8], where the case Φ ⁢ ( t ) = t p is treated.

How to cite

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Zecca, Gabriella. "On the Dirichlet Problem with Orlicz Boundary Data." Bollettino dell'Unione Matematica Italiana 10-B.3 (2007): 661-679. <http://eudml.org/doc/290386>.

@article{Zecca2007,
abstract = {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.},
author = {Zecca, Gabriella},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {661-679},
publisher = {Unione Matematica Italiana},
title = {On the Dirichlet Problem with Orlicz Boundary Data},
url = {http://eudml.org/doc/290386},
volume = {10-B},
year = {2007},
}

TY - JOUR
AU - Zecca, Gabriella
TI - On the Dirichlet Problem with Orlicz Boundary Data
JO - Bollettino dell'Unione Matematica Italiana
DA - 2007/10//
PB - Unione Matematica Italiana
VL - 10-B
IS - 3
SP - 661
EP - 679
AB - 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.
LA - eng
UR - http://eudml.org/doc/290386
ER -

References

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  8. KENIG, CARLOS E., Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, Conference Board of the Mathematical Sciences, Amer. Math. Soc.83 (1991). MR1282720DOI10.1090/cbms/083
  9. KERMAN, R. A. - TORCHINSKY, A., Integral inequalities with weights for the Hardy maximal function, Studia Math., 71 (1982), 277-284. Zbl0517.42030MR667316DOI10.4064/sm-71-3-277-284
  10. MATUSZEWSKA, W. - ORLICZ, W., On certain properties of W-functions, Bull. Acad. Polon. Sci., 8, 7 , (1960), 439-443. Zbl0101.09001MR126158
  11. MOSCARIELLO, G. - SBORDONE, C., a ∞ as a limit case of reverse - Hölder inequality when the exponent tends to 1, Ricerche Mat., XLIV, 1 (1995), 131-144. Zbl0920.26017MR1470190
  12. MUCKENHOUPT, B., Weighted norm inqualities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226. Zbl0236.26016MR293384DOI10.2307/1995882
  13. STEIN, E. M. - WEISS, G., An extension of a theorem of Marcinkiewicz and some of its applications, J. Math. Mech., 8 (1959), 263-264. Zbl0084.10801MR107163
  14. ZECCA, G., The unsolvability of the Dirichlet problem with L ⁢ ( log ⁡ L ) a boundary data, Rend. Acc. Sc. Fis. Mat. Napoli, 72 (2005), 71-80. Zbl1211.35090MR2449907

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