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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Stationary isothermic surfaces and uniformly dense domains
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by R. Magnanini, J. Prajapat and S. Sakaguchi PDF
Trans. Amer. Math. Soc. 358 (2006), 4821-4841 Request permission

Abstract:

We establish a relationship between stationary isothermic surfaces and uniformly dense domains. A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain $\Omega$ in the $N$-dimensional Euclidean space $\mathbb {R}^N$ is said to be uniformly dense in a surface $\Gamma \subset \mathbb {R}^N$ of codimension $1$ if, for every small $r>0,$ the volume of the intersection of $\Omega$ with a ball of radius $r$ and center $x$ does not depend on $x$ for $x\in \Gamma .$

We prove that the boundary of every uniformly dense domain which is bounded (or whose complement is bounded) must be a sphere. We then examine a uniformly dense domain with unbounded boundary $\partial \Omega$, and we show that the principal curvatures of $\partial \Omega$ satisfy certain identities.

The case in which the surface $\Gamma$ coincides with $\partial \Omega$ is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) if $N=2$, it must be either a circle or a straight line and (ii) if $N=3,$ it must be either a sphere, a spherical cylinder or a minimal surface. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface.

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Additional Information
  • R. Magnanini
  • Affiliation: Dipartimento di Matematica U. Dini, Università di Firenze, viale Morgagni 67/A, 50134 Firenze, Italy
  • Email: magnanin@math.unifi.it
  • J. Prajapat
  • Affiliation: Indian Statistical Institute, Stat-Math Unit, 8th Mile, Mysore Road, R.V.C.E. Post, Bangalore 560 059, India
  • Address at time of publication: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400 097, India
  • Email: jyotsna@isibang.ac.in
  • S. Sakaguchi
  • Affiliation: Department of Mathematics, Faculty of Science, Ehime University, 2-5 Bunkyo-cho, Matsuyama-shi, Ehime 790-8577 Japan
  • Email: sakaguch@dpc.ehime-u.ac.jp
  • Received by editor(s): August 11, 2004
  • Published electronically: April 11, 2006
  • Additional Notes: This research was partially supported by a Grant-in-Aid for Scientific Research (B) ($\sharp$ 12440042) and (B) ($\sharp$ 15340047) of the Japan Society for the Promotion of Science and by a Grant of the Italian MURST
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 358 (2006), 4821-4841
  • MSC (2000): Primary 35K05, 35K20; Secondary 53A10, 58J70
  • DOI: https://doi.org/10.1090/S0002-9947-06-04145-6
  • MathSciNet review: 2231874