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Book Review

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Book Information:

Authors: Luis A. Caffarelli and Xavier Cabré
Title: Fully nonlinear elliptic equations
Additional book information: Amer. Math. Soc. Colloq. Publ., vol. 43, Amer. Math. Soc., Providence, RI, 1995, vi + 104 pp., ISBN 0-8218-0437-5, $39.00$

Luis A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189–213. MR 1005611, DOI 10.2307/1971480

Luis Caffarelli, Elliptic second order equations, Rend. Sem. Mat. Fis. Milano 58 (1988), 253–284 (1990). MR 1069735, DOI 10.1007/BF02925245

Michael G. Crandall and Pierre-Louis Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), no. 1, 1–42. MR 690039, DOI 10.1090/S0002-9947-1983-0690039-8

Lawrence C. Evans, A convergence theorem for solutions of nonlinear second-order elliptic equations, Indiana Univ. Math. J. 27 (1978), no. 5, 875–887. MR 503721, DOI 10.1512/iumj.1978.27.27059

Lawrence C. Evans, On solving certain nonlinear partial differential equations by accretive operator methods, Israel J. Math. 36 (1980), no. 3-4, 225–247. MR 597451, DOI 10.1007/BF02762047

Lawrence C. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. Pure Appl. Math. 35 (1982), no. 3, 333–363. MR 649348, DOI 10.1002/cpa.3160350303

Lawrence C. Evans, Classical solutions of the Hamilton-Jacobi-Bellman equation for uniformly elliptic operators, Trans. Amer. Math. Soc. 275 (1983), no. 1, 245–255. MR 678347, DOI 10.1090/S0002-9947-1983-0678347-8

David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0

Hitoshi Ishii, On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs, Comm. Pure Appl. Math. 42 (1989), no. 1, 15–45. MR 973743, DOI 10.1002/cpa.3160420103

Hitoshi Ishii, Perron’s method for Hamilton-Jacobi equations, Duke Math. J. 55 (1987), no. 2, 369–384. MR 894587, DOI 10.1215/S0012-7094-87-05521-9

Robert Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech. Anal. 101 (1988), no. 1, 1–27. MR 920674, DOI 10.1007/BF00281780

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 3, 487–523, 670 (Russian). MR 661144

N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 1, 75–108 (Russian). MR 688919

N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Mathematics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the Russian by P. L. Buzytsky [P. L. Buzytskiĭ]. MR 901759, DOI 10.1007/978-94-010-9557-0

N. V. Krylov and M. V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure, Dokl. Akad. Nauk SSSR 245 (1979), no. 1, 18–20 (Russian). MR 525227

N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 161–175, 239 (Russian). MR 563790

P.-L. Lions, Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. I. The dynamic programming principle and applications, Comm. Partial Differential Equations 8 (1983), no. 10, 1101–1174. MR 709164, DOI 10.1080/03605308308820297

Review Information:

Reviewer: John Urbas
Affiliation: University of Bonn
Email: urbas@math.uni-bonn.de