The first eigenvalue of the  p ‐Laplace operator under powers of mean curvature flow | Zendy

Zhao Liang | Zendy

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The first eigenvalue of the p ‐Laplace operator under powers of mean curvature flow

Author(s) -

Zhao Liang

Publication year - 2013

Publication title -

mathematical methods in the applied sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.719

H-Index - 65

eISSN - 1099-1476

pISSN - 0170-4214

DOI - 10.1002/mma.2835

Subject(s) - mathematics , eigenvalues and eigenvectors , differentiable function , laplace transform , monotonic function , operator (biology) , constant (computer programming) , flow (mathematics) , almost everywhere , mathematical analysis , laplace operator , manifold (fluid mechanics) , curvature , pure mathematics , mathematical physics , combinatorics , geometry , quantum mechanics , physics , mechanical engineering , biochemistry , chemistry , repressor , computer science , transcription factor , engineering , gene , programming language

In this paper, we show the following main results. Let ( M n ,g(t)), t ∈ [0, T ), be a solution of the unnormalized H k − flow on a closed manifold, and λ 1, p ( t ) be the first eigenvalue of the p ‐Laplace operator. If there exists a nonnegative constant ε such thatH kh ij −H k + 1pg ij ≥ − ε g ijin M × [0, T ) andH k + 1 > pε in M × [0, T ),then λ 1, p ( t ) is increasing and the differentiable almost everywhere along the unnormalized H k − flow on [0, T ). At last, we discuss some useful monotonic quantities. Copyright © 2013 John Wiley & Sons, Ltd.

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