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The differentiability of solutions for elliptic equations which degenerate on part of the boundary of a convex domain

  • Jia-xin Song
  • Yi CaoEmail author
Article
  • 8 Downloads

Abstract

In this paper, we study the differentiability of solutions on the boundary for equations of type \({L_\lambda }u = \frac{{{\partial ^2}u}}{{\partial {x^2}}} + {\left| x \right|^{2\lambda }}\frac{{{\partial ^2}u}}{{\partial {y^2}}} = f\left( {x,y} \right)\) , where λ is an arbitrary positive number. By introducing a proper metric that is related to the elliptic operator Lλ, we prove the differentiability on the boundary when some well-posed boundary conditions are satisfied. The main diffculty is the construction of new barrier functions in this article.

Keywords

elliptic equations convex domain differentiability 

MR Subject Classification

35J70 35H20 

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Notes

Acknowledgments

The authors wish to express our sincere thanks to the referees for their careful reading and helpful comments

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Copyright information

© Editorial Committee of Applied Mathematics-A Journal of Chinese Universities and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Information ScienceShaanxi Normal UniversityXi’anChina

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