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Weak Formulation for Singular Diffusion Equation with Dynamic Boundary Condition

  • Ryota Nakayashiki
  • Ken ShirakawaEmail author
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 22)

Abstract

In this paper, we propose a weak formulation of the singular diffusion equation subject to the dynamic boundary condition. The weak formulation is based on a reformulation method by an evolution equation including the subdifferential of a governing convex energy. Under suitable assumptions, the principal results of this study are stated in forms of Main Theorems A and B, which are respectively to verify: the adequacy of the weak formulation; the common property between the weak solutions and those in regular problems of standard PDEs.

Keywords

Comparison principle Dynamic boundary condition Evolution equation Governing convex energy Mosco-convergence Singular diffusion equation 

AMS Subject Classification

??35K20 35K67 49J45 

Notes

Acknowledgements

This paper is dedicated to Professor Gianni Gilardi on the occasion of his 70th birthday. On a final note, we appreciate very much to the anonymous referee for taking great efforts to review our manuscript, and for giving us a lot of valuable comments and remarks. Ken Shirakawa is supported by Grant-in-Aid No. 16K05224, JSPS.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsGraduate School of Science, Chiba UniversityChibaJapan
  2. 2.Department of Mathematics, Faculty of EducationChiba UniversityChibaJapan

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