Linear elliptic equations with BMO coefficients

Menita Carozza; Gioconda Moscariello; Antonia Passarelli di Napoli

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1999)

  • Volume: 10, Issue: 1, page 17-23
  • ISSN: 1120-6330

Abstract

top
We prove an existence and uniqueness theorem for the Dirichlet problem for the equation div ⁢ a ⁢ x ⁢ ∇ ⁡ u = div ⁢ f in an open cube Ω ⊂ R N , when f belongs to some L p ⁢ Ω , with p close to 2. Here we assume that the coefficient a belongs to the space BMO( Ω ) of functions of bounded mean oscillation and verifies the condition a ⁢ x ≥ λ 0 > 0 for a.e. x ∈ Ω .

How to cite

top

Carozza, Menita, Moscariello, Gioconda, and Passarelli di Napoli, Antonia. "Linear elliptic equations with BMO coefficients." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 10.1 (1999): 17-23. <http://eudml.org/doc/252297>.

@article{Carozza1999,
abstract = {We prove an existence and uniqueness theorem for the Dirichlet problem for the equation \( \text\{div\} (a(x) \nabla u) = \text\{div\} f \) in an open cube \( \Omega \subset \mathbb\{R\}^\{N\} \), when \( f \) belongs to some \( L^\{p\} (\Omega) \), with \( p \) close to 2. Here we assume that the coefficient \( a \) belongs to the space BMO(\( \Omega \)) of functions of bounded mean oscillation and verifies the condition \( a(x) \ge \lambda\_\{0\} > 0 \) for a.e. \( x \in \Omega \).},
author = {Carozza, Menita, Moscariello, Gioconda, Passarelli di Napoli, Antonia},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Dirichlet problem; Existence and regularity; BMO-space},
language = {eng},
month = {3},
number = {1},
pages = {17-23},
publisher = {Accademia Nazionale dei Lincei},
title = {Linear elliptic equations with BMO coefficients},
url = {http://eudml.org/doc/252297},
volume = {10},
year = {1999},
}

TY - JOUR
AU - Carozza, Menita
AU - Moscariello, Gioconda
AU - Passarelli di Napoli, Antonia
TI - Linear elliptic equations with BMO coefficients
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1999/3//
PB - Accademia Nazionale dei Lincei
VL - 10
IS - 1
SP - 17
EP - 23
AB - We prove an existence and uniqueness theorem for the Dirichlet problem for the equation \( \text{div} (a(x) \nabla u) = \text{div} f \) in an open cube \( \Omega \subset \mathbb{R}^{N} \), when \( f \) belongs to some \( L^{p} (\Omega) \), with \( p \) close to 2. Here we assume that the coefficient \( a \) belongs to the space BMO(\( \Omega \)) of functions of bounded mean oscillation and verifies the condition \( a(x) \ge \lambda_{0} > 0 \) for a.e. \( x \in \Omega \).
LA - eng
KW - Dirichlet problem; Existence and regularity; BMO-space
UR - http://eudml.org/doc/252297
ER -

References

top
  1. Brezis, H. - Nirenberg, L., Degree theory and BMO; Part I: Compact manifolds with boundary. Selecta Math., 1 (2), 1995, 197-263. Zbl0852.58010MR1354598DOI10.1007/BF01671566
  2. Coifman, R. R. - Lions, P.L. - Meyer, Y. - Semmes, S., Compensated compactness and Hardy spaces. J. Math. Pures Appl., 1993, 247-286. Zbl0864.42009MR1225511
  3. Fiorenza, A. - Sbordone, C., Existence and uniqueness results of nonlinear equations with right hand side in L 1 . Studia Math., 127 (3), 1998, 223-231. Zbl0891.35039MR1489454
  4. Iwaniec, T. - Sbordone, C., Weak minima of variational integrals. J. Reine Angew. Math., 454, 1994, 143-161. Zbl0802.35016MR1288682DOI10.1515/crll.1994.454.143
  5. Iwaniec, T. - Verde, A., A study of Jacobians in Hardy-Orlicz spaces. Proc. Royal Soc. of Edinburgh, to appear. Zbl0954.46018
  6. Meyers, N., An L p -estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Sc. Norm. Pisa, 17, 1963, 189-206. Zbl0127.31904MR159110
  7. Miyachi, A., H p spaces over open subset of R n . Studia Math., XCV, 1990, 204-228. Zbl0716.42017MR1060724
  8. Murat, F., Compacité par compensation. Ann. Sc. Normale Sup. Pisa, 5, 1978, 489-507. Zbl0399.46022MR506997
  9. Tartar, L., Compensated compactness and applications to partial differential equations in Nonlinear Analysis and Mechanics. Heriot Watt Symposium, Research Notes in Math., Pitman, London, 39, 1979, 136-212. Zbl0437.35004MR584398

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.