Abstract
We use similar functional analytic methods to solve (a) a fully nonlinear second order elliptic equation, (b) a Hamilton-Jacobi equation, and (c) a functional/partial differential equation from plasma physics. The technique in each case is to approximate by the solutions of simpler problems, and then to pass to limits using a modification of G. Minty’s device to the spaceL ∞.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. Barbu,Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976.
S. H. Benton, Jr.,The Hamilton Jacobi Equation: A Global Approach, Academic Press, New York, 1977.
J.-M. Bony,Principe du maximum dans les espaces de Sobolev, C. R. Acad. Sci. Paris265 (1967), 333–336.
F. Browder,Variational boundary value problems for quasilinear elliptic equations of arbitrary order, Proc. Nat. Acad. Sci. U.S.A.50 (1963), 31–37.
C. Burch,A semigroup treatment of the Hamilton-Jacobi equation in several space variables, J. Differential Equations23 (1977), 107–124.
M. G. Crandall,An introduction to evolution governed by accretive operators, Proceedings of the International Symposium on Dynamical Systems, Brown University, Providence, R. I., August 1974.
L. C. Evans,A convergence theorem for solutions of nonlinear second order elliptic equations, Indiana Univ. Math. J.27 (1978), 875–887.
L. C. Evans,Application of nonlinear semigroup theory to certain partial differential equations, inNonlinear Evolution Equations (M. G. Crandall, ed.), Academic Press, New York, 1978.
L. C. Evans and A. Friedman,Optimal stochastic switching and the Dirichlet problem for the Bellman equation, Trans. Amer. Math. Soc.253 (1979), 365–389.
W. H. Fleming,The Cauchy problem for degenerate parabolic equations, J. Math. Mech.13 (1964), 987–1008.
W. H. Fleming and R. W. Rishel,Deterministic and Stochastic Optimal Control, Springer, New York, 1975.
A. Friedman,Differential Games, Wiley-Interscience, New York, 1971.
A. Friedman,The Cauchy problem for first order partial differential equations, Indiana Univ. Math. J.23 (1973), 27–40.
H. Grad, P. N. Hu and D. C. Stevens,Proc. Nat. Acad. Sci. U.S.A. 72 (1975), 3789–3793.
S. N. Kruzkov,First order quasilinear equations in several independent variables, Math. USSR-Sb.10 (1970), 217–243.
J. L. Lions,Quelques méthods de résolution des problèmes aux limites non-linéaires, Dunod, Paris, 1969.
P. L. Lions,Résolution des problèmes de Bellman-Dirichlet: une méthode analytique I, to appear.
G. J. Minty,Monotone (non-linear) operators in Hilbert space, Duke Math. J.29 (1962), 341–346.
J. Mossino,Sur certaines inéquations quasi-variationnelles apparaissant en physique, C. R. Acad. Sci. Paris282 (1976), 187–190.
J. Mossino,Application des inéquations quasi-variationelles a quelques problèmes non linéaires de la physique des plasmas, Israel J. Math.30, (1978), 14–50.
J. Mossino and J.-P. Zolesio,Solution variationnelle d’un problème non linéaire de la physique des plasmas, C. R. Acad. Sci. Paris285 (1977), 1033–1036.
K. Sato,On the generators of non-negative contraction semi-groups in Banach lattices, J. Math. Soc. Japan20 (1968), 423–436.
E. Sinestrari,Accretive differential operators, Boll. Un. Mat. Ital.13 (1976), 19–31.
M. B. Tamburro,The evolution operator solution of the Cauchy problem for the Hamilton-Jacobi equation, Israel J. Math.26 (1977), 232–264.
Author information
Authors and Affiliations
Additional information
Alfred P. Sloan fellow 1979–1981.
Supported in part by NSF grant MCS 77-01952.
Rights and permissions
About this article
Cite this article
Evans, L.C. On solving certain nonlinear partial differential equations by accretive operator methods. Israel J. Math. 36, 225–247 (1980). https://doi.org/10.1007/BF02762047
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02762047