Elliptic Equations in Divergence Form with Partially BMO Coefficients

By Hongjie Dong, Doyoon Kim. 2008

127 citations recorded by the source corpus.

The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients aij are assumed to be only measurable in one direction and have locally small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume that aij have small BMO semi-norms in a neighborhood of the boundary of the domain. We give a unified approach of both the Dirichlet boundary problem and the conormal derivative problem. We also investigate elliptic equations in Sobolev spaces with mixed norms under the same assumptions on the coefficients.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/753a1939525584affb43430f252ce003d77ae33d

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In Applied Mathematics

1. Hongjie Dong
2. Doyoon Kim
3. Sun-Sig Byun
4. Lihe Wang
5. Zhongwei Shen
6. N. Phuc
7. Seungjin Ryu
8. Giuseppe Mingione

In Computational Theory and Mathematics

1. Hongjie Dong
2. Doyoon Kim
3. Sun-Sig Byun
4. Lihe Wang
5. Zhongwei Shen
6. N. Phuc
7. Seungjin Ryu
8. Giuseppe Mingione

In Mathematical Physics

1. Sun-Sig Byun
2. Hongjie Dong
3. Lihe Wang
4. Doyoon Kim
5. Zhongwei Shen
6. N. Phuc
7. Giuseppe Mingione
8. Jihoon Ok

Citations in the corpus

• Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains (2011, methodology): “In the half space case, compared to previous results in [22, 23, 14] for second order equations, one of the virtues of our results is that on each small cylinder centered on the boundary, we do not require the measurable direction to be exactly perpendicular to the boundary, but allow it to be suffi”
• On the Lp-Solvability of Higher Order Parabolic and Elliptic Systems with BMO Coefficients (2010, methodology): “When dealing with equations and systems on a half space or on a bounded domain in [22, 23, 12, 14], the authors took full advantage of the facts that the coefficients are allowed to be merely measurable in one spatial direction”
• Elliptic equations with measurable coefficients in Reifenberg domains (2010, result): “In what concerns divergence and non-divergence form elliptic and parabolic operators whose coefficients are measurable in one variable and VMO with respect to the others, we would like to mention a series of recent works [11,12,17–19] where D.”
• Boundary Estimates in Elliptic Homogenization (2015, background): “We remark that theW 1,p estimates (local or global) for operators with nonsmooth coefficients in nonsmooth domains are of interest in their own rights and have been studied extensively in recent years (see [11, 45, 8, 39, 9, 34, 10, 41, 13, 29, 16, 18] and their references).”
• Lp solvability of divergence type parabolic and elliptic systems with partially BMO coefficients (2011, background): “For proofs, we refer the interested reader to [9].”
• On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights (2016, background): “Concerning the second order equations/systems case, see, for instance, [17, 19, 21, 8, 6] and references therein.”