On the definition of viscosity solutions for parabolic equations

Author:
Petri Juutinen

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

Published electronically:
February 15, 2001

MathSciNet review:
1840092

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[C]**Crandall, M. G.,*Viscosity solutions: a primer*, Viscosity solutions and applications (Montecatini Terme, 1995), Lecture Notes in Math., 1660, Springer, Berlin (1997), 1-43. MR**98g:35034****[CIL]**Crandall, M. G., H. Ishii, and P-.L. Lions,*User's guide to viscosity solutions of second order partial differential equations*, Bull. Amer. Math. Soc.**27**(1992), 1-67. MR**92j:35050****[DiB]**DiBenedetto, E.,*Degenerate parabolic equations*, Springer-Verlag, New York (1993). MR**94h:35130****[F]**Friedman, A.,*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1964). MR**31:6062****[IS]**Ishii, H., and P. E. Souganidis,*Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor*, Tôhoku Math. J.**47**(1995), 227-250. MR**96e:35069****[JLM]**Juutinen, P., P. Lindqvist, and J. Manfredi,*On the equivalence of viscosity solutions and weak solutions for a quasilinear equation*, manuscript (2000).**[KL]**Kilpeläinen, T., and P. Lindqvist,*On the Dirichlet boundary value problem for a degenerate parabolic equation*, SIAM J. Math. Anal.**27**(1996), 661-683. MR**97b:35118****[SZ]**Sternberg, P., and W. P. Ziemer,*Generalized motion by curvature with a Dirichlet condition*, J. Differential Equations**114**(1994), 580-600. MR**96a:35096****[W]**Wu, Y.,*Absolute minimizers in Finsler metrics*, Ph.D. dissertation, UC Berkeley (1995).

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35K55,
35D99,
35B40

Retrieve articles in all journals with MSC (2000): 35K55, 35D99, 35B40

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

DOI:
https://doi.org/10.1090/S0002-9939-01-05889-0

Keywords:
Viscosity solutions,
parabolic equations

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

Article copyright:
© Copyright 2001
American Mathematical Society