On the range of elliptic operators discontinuous at one point

Cristina Giannotti

Bollettino dell'Unione Matematica Italiana (2002)

  • Volume: 5-B, Issue: 1, page 123-129
  • ISSN: 0392-4041

Abstract

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Let L be a second order, uniformly elliptic, non variational operator with coefficients which are bounded and measurable in R d ( d ≥ 3 ) and continuous in R d ∖ 0 . Then, if Ω ⊂ R d is a bounded domain, we prove that L ⁢ W 2 , p ⁢ Ω is dense in L p ⁢ Ω for any p ∈ 1 , d / 2 .

How to cite

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Giannotti, Cristina. "On the range of elliptic operators discontinuous at one point." Bollettino dell'Unione Matematica Italiana 5-B.1 (2002): 123-129. <http://eudml.org/doc/194662>.

@article{Giannotti2002,
abstract = {Let $L$ be a second order, uniformly elliptic, non variational operator with coefficients which are bounded and measurable in $\mathbb\{R\}^\{d\}$ ($d\geq3$) and continuous in $\mathbb\{R\}^\{d\} \setminus \\{0\\}$. Then, if $\Omega\subset \mathbb\{R\}^\{d\}$ is a bounded domain, we prove that $L(W^\{2, p \}(\Omega))$ is dense in $L^\{p\}(\Omega)$ for any $p\in(1, d/2 ]$.},
author = {Giannotti, Cristina},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {2},
number = {1},
pages = {123-129},
publisher = {Unione Matematica Italiana},
title = {On the range of elliptic operators discontinuous at one point},
url = {http://eudml.org/doc/194662},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Giannotti, Cristina
TI - On the range of elliptic operators discontinuous at one point
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/2//
PB - Unione Matematica Italiana
VL - 5-B
IS - 1
SP - 123
EP - 129
AB - Let $L$ be a second order, uniformly elliptic, non variational operator with coefficients which are bounded and measurable in $\mathbb{R}^{d}$ ($d\geq3$) and continuous in $\mathbb{R}^{d} \setminus \{0\}$. Then, if $\Omega\subset \mathbb{R}^{d}$ is a bounded domain, we prove that $L(W^{2, p }(\Omega))$ is dense in $L^{p}(\Omega)$ for any $p\in(1, d/2 ]$.
LA - eng
UR - http://eudml.org/doc/194662
ER -

References

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  1. ARENA, O., On the range of Ural'tseva's Axially symmetric Operator in Sobolev Spaces, Partial Differential Equations (P. Marcellini, G. Talenti, E. Vesentini Eds.) Dekker (1996). Zbl0867.35029
  2. GILBARG, D.- SERRIN, J., On isolated singularities of solutions of second order elliptic equations, J. Anal. Math., 4 (1955-56), 309-340. Zbl0071.09701MR81416
  3. GILBARG, D.- TRUDINGER, N. S., Elliptic Partial Differential Equations of Second Order, Springer (1983). Zbl0562.35001MR737190
  4. LADYZHENSKAYA, O. A.- URAL'TSEVA, N. N., Linear and Quasilinear Elliptic Equations, A.P. (1968). Zbl0164.13002MR244627
  5. MANSELLI, P., On the range of elliptic, second order, nonvariational operators in Sobolev spaces, Annali Mat. pura e appl., (IV), Vol. CLXXVIII (2000), 67-80. Zbl1096.47518MR1849379
  6. NEČAS, J., Les Méthodes Directes en Théorie des Équations Elliptiques, MassonParis1967. MR227584
  7. PUCCI, C., Operatori ellittici estremanti, Annali di Matematica Pura ed Applicata (IV), Vol. LXXII (1966), 141-170. Zbl0154.12402MR208150
  8. URAL'TSEVA, N. N., Impossibility of W 2 , p bounds for multidimensional elliptic operators with discontinuous coefficients, L.O.M.I., 5 (1967), 250-254. Zbl0186.43006MR226179

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