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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The infinity Laplacian, Aronsson’s equation and their generalizations
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by E. N. Barron, L. C. Evans and R. Jensen PDF
Trans. Amer. Math. Soc. 360 (2008), 77-101 Request permission

Abstract:

The infinity Laplace equation $\Delta _\infty u=0$ arose originally as a sort of Euler–Lagrange equation governing the absolute minimizer for the $L^{\infty }$ variational problem of minimizing the functional $\text {ess-sup}_{U}|Du|.$ The more general functional $\text {ess-sup}_{U}F(x,u,Du)$ leads similarly to the so-called Aronsson equation $A_F[u]=0.$ In this paper we show that these PDE operators and various interesting generalizations also appear in several other contexts seemingly quite unrelated to $L^{\infty }$ variational problems, including two-person game theory with random order of play, rapid switching of states in control problems, etc. The resulting equations can be parabolic and inhomogeneous, equation types precluded in conventional $L^{\infty }$ variational problems.
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Additional Information
  • E. N. Barron
  • Affiliation: Department of Mathematics and Statistics, Loyola University of Chicago, Chicago, Illinois 60626
  • MR Author ID: 31685
  • Email: ebarron@luc.edu
  • L. C. Evans
  • Affiliation: Department of Mathematics, University of California, Berkeley, Berkeley, California 94720
  • Email: evans@math.berkeley.edu
  • R. Jensen
  • Affiliation: Department of Mathematics and Statistics, Loyola University of Chicago, Chicago, Illinois 60626
  • MR Author ID: 205502
  • Email: rjensen@luc.edu
  • Received by editor(s): July 15, 2005
  • Published electronically: July 25, 2007
  • Additional Notes: The first and third authors were supported in part by NSF Grant DMS-0200169
    The second author was supported in part by NSF Grant DMS-0500452.
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 77-101
  • MSC (2000): Primary 35C99, 35J60; Secondary 49L20, 49L25
  • DOI: https://doi.org/10.1090/S0002-9947-07-04338-3
  • MathSciNet review: 2341994