Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. D. Aleksandrov, Almost everywhere existence of the second differential of a convex function and some properties of convex functions, Leningrad University Annals (Mathematical Series) 37 (1939), 3–35 (in Russian).
M. Bardi, Some applications of viscosity solutions to optimal control and differential games, this volume.
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Springer-Verlag, New York, 1994.
G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping-time problems, Modèl. Math. et Anal. Num. 21 (1987), 557–579.
-, Exit time problems in optimal control and the vanishing viscosity method, SIAM J. Control Optim. 26 (1988), 1133–1148.
G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymp. Anal. 4 (1989), 271–283.
X. Cabré and L. Caffarelli, Fully Nonlinear Elliptic Equations, Amer. Math. Society, Providence, 1995.
L. Caffarelli, M. Crandall, M. Kocan and A. Święch, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49 (1996), 365–397.
F. H. Clarke, Y. S. Ledyaev, R. J. Stern, and P. R. Wolenksi, Qualitative properties of trajectories of control systems—a survey, Journal of Dynamical and Control Systems 1 (1995), 1–48.
M. G. Crandall, Quadratic forms, semidifferentials and viscosity solutions of fully nonlinear elliptic equations, Ann. I.H.P. Anal. Non. Lin. 6 (1989), 419–435.
M. G. Crandall and H. Ishii, The maximum principle for semicontinuous functions, Diff. and Int. Equations 3 (1990), 1001–1014.
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. (N.S.) 27 (1992), 1–67.
M. G. Crandall, M. Kocan, P. Soravia and A. Święch: On the equivalence of various weak notions of solutions of elliptic pdes with measurable ingredients, in Progress in elliptic and parabolic partial differential equations, Alvino et. al. eds, Pitman Research Notes 350, Addison Wesley Longman, 1996, p 136–162.
M. G. Crandall and P. L. Lions, Condition d’unicité pour les solutions generalisées des équations de Hamilton-Jacobi du premier ordre, C. R. Acad. Sci. Paris 292 (1981), 183–186.
-, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1–42.
M. G. Crandall, P. L. Lions and L. C. Evans, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282 (1984), 487–502.
G. Dong, Nonlinear Partial Differential Equations of Second Order, Translations of Mathematical Monographs 95, American Mathematical Society, Providence, 1994.
L. C. Evans, A convergence theorem for solutions of nonlinear second order elliptic equations, Indiana Univ. J. 27 (1978), 875–887.
-, On solving certain nonlinear differential equations by accretive operator methods, Israel J. Math. 36 (1980), 225–247.
L. C. Evans, Regularity for fully nonlinear elliptic equations and motion by mean curvature, this volume.
L. C. Evans, and R. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1992.
W. H. Fleming and H. Mete Soner, Controlled Markov Processes and Viscosity Solutions, Applications of Mathematics 25, Springer-Verlag, New York, 1993.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second order, 2 nd Edition, Springer-Verlag, New York, 1983.
H. Ishii, Perron’s method for Hamilton-Jacobi equations, Duke Math. J. 55 (1987), 369–384.
-, On uniqueness and existence of viscosity solutions of fully non-linear second order elliptic PDE’s, Comm. Pure Appl. Math. 42 (1989), 14–45.
H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second order elliptic partial differential equations, J. Diff. Equa. 83 (1990), 26–78.
R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rat. Mech. Anal. 101 (1988), 1–27.
-, Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential equations, Indiana U. Math. J. 38 (1989), 629–667.
-, Uniqueness of Lipschitz extensions—Minimizing the sup norm or the gradient, Arch. Rat. Mech. Anal 123 (1993), 51–74.
R. Jensen, P. L. Lions and P. E. Souganidis, A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations, Proc. AMS 102 (1988), 975–978.
P. L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 1: The dynamic programming principle and applications and Part 2: Viscosity solutions and uniqueness, Comm. P. D. E. 8 (1983), 1101–1174 and 1229–1276.
J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math 55 (1986), 257–266.
K. Miller, Barriers on cones for uniformly elliptic equations, Ann. di Mat. Pura Appl. LXXVI (1967), 93–106.
M. Soner, Controlled Markov processes, viscosity solutions and applications to mathematical finance, this volume.
P. E. Souganidis, Front Propagation: Theory and applications, this volume.
A. Subbotin, Solutions of First-order PDEs. The Dynamical Optimazation Perspective, Birkhauser, Boston, 1995.
A. Święch, W 1,p-interior estimates for solutions of fully nonlinear, uniformly elliptic equations, preprint.
N. S. Trudinger, Comparison principles and pointwise estimates for viscosity solutions of second order elliptic equations, Rev. Mat. Iberoamericana 4 (1988), 453–468.
-, Hölder gradient estimates for fully nonlinear elliptic equations, Proc. Roy. Soc. Edinburgh, Sect. A 108 (1988), 57–65.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag
About this chapter
Cite this chapter
Crandall, M.G. (1997). Viscosity solutions: A primer. In: Dolcetta, I.C., Lions, P.L. (eds) Viscosity Solutions and Applications. Lecture Notes in Mathematics, vol 1660. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094294
Download citation
DOI: https://doi.org/10.1007/BFb0094294
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-62910-8
Online ISBN: 978-3-540-69043-6
eBook Packages: Springer Book Archive