Hölder continuity of solutions to a degenerate elliptic equation in the presence of the Lavrentiev phenomenon

We study a second-order elliptic equation for which the Dirichlet problem can be posed in a nonunique way due to the so-called Lavrentiev phenomenon. In the corresponding weighted Sobolev space smooth functions are not dense, which leads to the existence of W - solutions and H - solutions. For H - s...

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Bibliographic Details
Published in:Applicable analysis 2014-10, Vol.93 (10), p.2057-2075
Main Authors: Alkhutov, Yu. A., Zhikov, V.V.
Format: Article
Language:English
Subjects:
Continuity
degenerate elliptic equations
Dirichlet problem
Euclidean space
Hölder continuity
Lavrentiev phenomenon
Mathematical analysis
Partial differential equations
Sobolev space
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Summary:We study a second-order elliptic equation for which the Dirichlet problem can be posed in a nonunique way due to the so-called Lavrentiev phenomenon. In the corresponding weighted Sobolev space smooth functions are not dense, which leads to the existence of W - solutions and H - solutions. For H - solutions, we establish the Hölder continuity. We also discuss this question for W - solutions, for which the situation is more complicated.
ISSN:0003-6811
1563-504X
DOI:10.1080/00036811.2014.936404