A Harnack inequality for a transmission problem with p(x)-Laplacian

ABSTRACT

In this paper, we establish a Harnack type inequality for p(x)-harmonic functions when the exponent p(x) makes a jump at a hyperplane and satisfies log-condition otherwise. We demonstrate that in this situation the Harnack inequality in the classical form is not valid. The form of the Harnack inequality we obtain is weaker than the classical one but still powerful enough to imply the Hölder continuity.

KEYWORDS:

• Harnack inequality
• transmission problem
• variable exponent
• p(x)-Laplacian

AMS Subject Classifications:

• 35J15
• 35J92

Acknowledgements

The authors thank the unanymous referee for his careful reading of this manuscript and valuable remarks.

Notes

No potential conflict of interest was reported by the authors.

Dedicated to the memory of Vasily V. Zhikov, who was a great Mathematician, Friend and Teacher.

Additional information

Funding

This work was supported by the Russian Foundation for Basic Research [grant number 15-01-00471]; the Ministry of Education and Science of the Russian Federation [grant number 1.3270.2017/4.6].