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Estimates for viscosity solutions of parabolic equations with Dirichlet boundary conditions


Author: G. Gripenberg
Journal: Proc. Amer. Math. Soc. 130 (2002), 3651-3660
MSC (2000): Primary 35K55, 35K65, 35K20
DOI: https://doi.org/10.1090/S0002-9939-02-06580-2
Published electronically: May 1, 2002
MathSciNet review: 1920045
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Abstract | References | Similar Articles | Additional Information

Abstract: It is shown how one can get upper bounds for $\lvert u-v \rvert$when $u$ and $v$ are the (viscosity) solutions of

\begin{displaymath}u_t - \alpha(D_x u) \Delta_x u = 0\quad\text{and}\quad v_t - \beta(D_x v) \Delta_x v = 0, \end{displaymath}

respectively, in $(0,\infty)\times \Omega$ with Dirichlet boundary conditions. Similar results are obtained for some other parabolic equations as well, including certain equations in divergence form.


References [Enhancements On Off] (What's this?)

  • 1. B. Cockburn, G. Gripenberg, and S-O. Londen.
    Continuous dependence on the nonlinearity of viscosity solutions of parabolic equations.
    J. Differential Equations, 170:180-187, 2001. CMP 2001:08
  • 2. M. G. Crandall.
    Viscosity solutions: a primer.
    In Viscosity solutions and applications (Montecatini Terme, 1995), volume 1660 of Lecture Notes in Math., pages 1-43. Springer, Berlin, 1997. MR 98g:35034
  • 3. M.G. Crandall, H. Ishii, and P.L. Lions.
    User's guide to viscosity solutions of second order partial differential equations.
    Bull. Amer. Math. Soc., 27:1-67, 1992. MR 92j:35050
  • 4. D. Nunziante.
    Existence and uniqueness of unbounded viscosity solutions of parabolic equations with discontinuous time-dependence.
    Nonlinear Anal., 18(11):1033-1062, 1992. MR 93f:35125
  • 5. T. Rockafellar.
    Convex analysis.
    Princeton University Press, Princeton, N.J., 1970. MR 43:445

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Additional Information

G. Gripenberg
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Finland
Email: gustaf.gripenberg@hut.fi

DOI: https://doi.org/10.1090/S0002-9939-02-06580-2
Keywords: Viscosity solution, parabolic, dependence on data, Dirichlet boundary condition
Received by editor(s): July 23, 2001
Published electronically: May 1, 2002
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2002 American Mathematical Society

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