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Eikonal equations in metric spaces


Authors: Yoshikazu Giga, Nao Hamamuki and Atsushi Nakayasu
Journal: Trans. Amer. Math. Soc. 367 (2015), 49-66
MSC (2010): Primary 35D40; Secondary 35F30, 49L25
DOI: https://doi.org/10.1090/S0002-9947-2014-05893-5
Published electronically: July 16, 2014
MathSciNet review: 3271253
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Abstract: A new notion of a viscosity solution for Eikonal equations in a general metric space is introduced. A comparison principle is established. The existence of a unique solution is shown by constructing a value function of the corresponding optimal control theory. The theory applies to infinite dimensional setting as well as topological networks, surfaces with singularities.


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Additional Information

Yoshikazu Giga
Affiliation: Graduate School of Mathematical Science, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8914, Japan
Email: labgiga@ms.u-tokyo.ac.jp

Nao Hamamuki
Affiliation: Graduate School of Mathematical Science, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8914, Japan
Address at time of publication: Faculty of Education and Integrated Arts and Sciences, Waseda University, 1-6-1 Nishi-Waseda, Shinjuku-ku, Tokyo 169-8050, Japan
Email: hnao@ms.u-tokyo.ac.jp

Atsushi Nakayasu
Affiliation: Graduate School of Mathematical Science, University of Tokyo, Komaba 3-8-1, Meguro-ku, Tokyo 153-8914, Japan
Email: ankys@ms.u-tokyo.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2014-05893-5
Keywords: Eikonal equation, metric space, metric viscosity solutions
Received by editor(s): December 17, 2011
Received by editor(s) in revised form: May 14, 2012
Published electronically: July 16, 2014
Additional Notes: The work of the first author was partly supported by a Grant-in-Aid for Scientific Research (S) No. 21224001 and (A) No. 23244015, Japan Society for the Promotion of Science (JSPS)
The work of the second author was supported by a Grant-in-Aid for JSPS Fellows No. 23-4365
The work of the third author was supported by a Grant-in-Aid for JSPS Fellows No. 25-7077 and the Program for Leading Graduate Schools, MEXT, Japan.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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