On the definition of viscosity solutions for parabolic equations
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- by Petri Juutinen PDF
- Proc. Amer. Math. Soc. 129 (2001), 2907-2911 Request permission
Abstract:
In this short note we suggest a refinement for the definition of viscosity solutions for parabolic equations. The new version of the definition is equivalent to the usual one and it better adapts to the properties of parabolic equations. The basic idea is to determine the admissibility of a test function based on its behavior prior to the given moment of time and ignore what happens at times after that.References
- Michael G. Crandall, Viscosity solutions: a primer, Viscosity solutions and applications (Montecatini Terme, 1995) Lecture Notes in Math., vol. 1660, Springer, Berlin, 1997, pp. 1–43. MR 1462699, DOI 10.1007/BFb0094294
- Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1–67. MR 1118699, DOI 10.1090/S0273-0979-1992-00266-5
- Emmanuele DiBenedetto, Degenerate parabolic equations, Universitext, Springer-Verlag, New York, 1993. MR 1230384, DOI 10.1007/978-1-4612-0895-2
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
- Hitoshi Ishii and Panagiotis Souganidis, Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor, Tohoku Math. J. (2) 47 (1995), no. 2, 227–250. MR 1329522, DOI 10.2748/tmj/1178225593
- Juutinen, P., P. Lindqvist, and J. Manfredi, On the equivalence of viscosity solutions and weak solutions for a quasilinear equation, manuscript (2000).
- T. Kilpeläinen and P. Lindqvist, On the Dirichlet boundary value problem for a degenerate parabolic equation, SIAM J. Math. Anal. 27 (1996), no. 3, 661–683. MR 1382827, DOI 10.1137/0527036
- Peter Sternberg and William P. Ziemer, Generalized motion by curvature with a Dirichlet condition, J. Differential Equations 114 (1994), no. 2, 580–600. MR 1303041, DOI 10.1006/jdeq.1994.1162
- Wu, Y., Absolute minimizers in Finsler metrics, Ph.D. dissertation, UC Berkeley (1995).
Additional Information
- Petri Juutinen
- Affiliation: Department of Mathematics, University of Jyväskylä, P.O. Box 35 (MaD), FIN-40351, Jyväskylä, Finland
- Email: peanju@math.jyu.fi
- Received by editor(s): August 23, 1999
- Received by editor(s) in revised form: February 2, 2000
- Published electronically: February 15, 2001
- Communicated by: Albert Baernstein II
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2907-2911
- MSC (2000): Primary 35K55, 35D99; Secondary 35B40
- DOI: https://doi.org/10.1090/S0002-9939-01-05889-0
- MathSciNet review: 1840092