Observations on W 1 , p estimates for divergence elliptic equations with VMO coefficients

P. Auscher; M. Qafsaoui

Bollettino dell'Unione Matematica Italiana (2002)

  • Volume: 5-B, Issue: 2, page 487-509
  • ISSN: 0392-4041

Abstract

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In this paper, we make some observations on the work of Di Fazio concerning W 1 , p estimates, 1 < p < ∞ , for solutions of elliptic equations div ⁢ A ⁢ ∇ ⁡ u = div ⁢ f , on a domain Ω with Dirichlet data 0 whenever A ∈ V ⁢ M ⁢ O ⁢ Ω and f ∈ L p ⁢ Ω . We weaken the assumptions allowing real and complex non-symmetric operators and C 1 boundary. We also consider the corresponding inhomogeneous Neumann problem for which we prove the similar result. The main tool is an appropriate representation for the Green (and Neumann) function on the upper half space. We propose two such representations.

How to cite

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Auscher, P., and Qafsaoui, M.. "Observations on $W^{1,p}$ estimates for divergence elliptic equations with VMO coefficients." Bollettino dell'Unione Matematica Italiana 5-B.2 (2002): 487-509. <http://eudml.org/doc/195627>.

@article{Auscher2002,
abstract = {In this paper, we make some observations on the work of Di Fazio concerning $W^\{1,p\}$ estimates, $1< p<\infty$, for solutions of elliptic equations $\text\{div\} \, A \nabla u = \text\{div\} f$ , on a domain $\Omega$ with Dirichlet data $0$ whenever $A \in VMO(\Omega)$ and $f \in L^\{p\} (\Omega)$. We weaken the assumptions allowing real and complex non-symmetric operators and $C^1$ boundary. We also consider the corresponding inhomogeneous Neumann problem for which we prove the similar result. The main tool is an appropriate representation for the Green (and Neumann) function on the upper half space. We propose two such representations.},
author = {Auscher, P., Qafsaoui, M.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {487-509},
publisher = {Unione Matematica Italiana},
title = {Observations on $W^\{1,p\}$ estimates for divergence elliptic equations with VMO coefficients},
url = {http://eudml.org/doc/195627},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Auscher, P.
AU - Qafsaoui, M.
TI - Observations on $W^{1,p}$ estimates for divergence elliptic equations with VMO coefficients
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/6//
PB - Unione Matematica Italiana
VL - 5-B
IS - 2
SP - 487
EP - 509
AB - In this paper, we make some observations on the work of Di Fazio concerning $W^{1,p}$ estimates, $1< p<\infty$, for solutions of elliptic equations $\text{div} \, A \nabla u = \text{div} f$ , on a domain $\Omega$ with Dirichlet data $0$ whenever $A \in VMO(\Omega)$ and $f \in L^{p} (\Omega)$. We weaken the assumptions allowing real and complex non-symmetric operators and $C^1$ boundary. We also consider the corresponding inhomogeneous Neumann problem for which we prove the similar result. The main tool is an appropriate representation for the Green (and Neumann) function on the upper half space. We propose two such representations.
LA - eng
UR - http://eudml.org/doc/195627
ER -

References

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