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Second order parabolic differential equations. (English) Zbl 0884.35001
Singapore: World Scientific. xi, 439 p. (1996).
The book provides an essentially self-contained exposition of the theory of second order parabolic partial differential equations. Two initial-boundary value problems in bounded (not necessarily cylindrical) domains are studied: the Cauchy-Dirichlet problem and the oblique derivative problem. The main aim is to prove existence, uniqueness, and regularity results for linear and nonlinear uniformly parabolic equations. Parabolic Monge-Ampère equations are also considered in the last chapter. The author uses methods that apply to single equations but they yield strong a priori estimates that are generally not available for systems of equations. The maximum principle plays an important role, and the use of integral representation formulae is avoided. Although a single point of view is chosen for most topics, the book contains a lot of material that will be very useful for specialists. The most closely related monographs seem to be D. Gilbarg and N. S. Trudinger [Elliptic partial differential equations of second order (Grundlehren der Math. Wissenschaften 224, Springer, Berlin) (1983; Zbl 0562.35001)] and O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’tseva [Linear and quasilinear equations of parabolic type (Translations of Math. Monographs 23, Am. Math. Soc., Providence R.I.) (1968; Zbl 0174.15403)]. Lieberman’s book might be viewed as a parabolic analog of the former or an updated analog of the latter.

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35K20 Initial-boundary value problems for second-order parabolic equations
35Bxx Qualitative properties of solutions to partial differential equations
35K55 Nonlinear parabolic equations
35K10 Second-order parabolic equations