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

Abstract

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.

Keywords:

• degenerate elliptic equations
• Hölder continuity
• Lavrentiev phenomenon

AMS Subject Classifications:

• 35J15
• 35J70

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Erratum

Acknowledgements

We also thank Mikhail Surnachev for careful reading of the manuscript.

Notes

Dedicated to Prof. A. Pankov on the occasion of his 65th birthday.

This work was supported by RFBR (research project 12-01-00058-a) and carried out in the framework of the government job of the Ministry of Science and Education of the Russian Federation (job No 2014/13, project code 3037). Results of Paragraph 2 were obtained with the support of the Russian Science Fund, project 14-11-00398.