Nonlinear elliptic equations of the second order (Q2808359)

The textbook under review presents a detailed discussion on the Dirichlet problems for quasilinear and fully nonlinear elliptic PDEs of second order, with an emphasis on two important equations closely related to the geometry, the mean curvature equation and the Monge-Ampère equation. More precisely, the author provides basic results regarding quasilinear uniformly elliptic equations in arbitrary domains, mean curvature equation over domains with nonnegative boundary mean curvature, fully nonlinear uniformly elliptic equations in arbitrary domains and Monge-Ampère equation in uniformly convex domains.NEWLINENEWLINENEWLINEThe book gives a user-friendly introduction to the theory of nonlinear elliptic equations with special attention given to basic results and the most important techniques. Rather than presenting the topics in their full generality, it aims at providing self-contained, clear, and somewhat ``elementary'' proofs for results in the above mentioned special cases.NEWLINENEWLINENEWLINEThe author starts with reviewing briefly three basic topics from the theory of linear elliptic PDEs: the maximum principle, Krylov-Safonov's Harnack inequality and the Schauder theory.NEWLINENEWLINENEWLINEChapter 2 discusses general quasilinear elliptic equations. Various a priori estimates are derived for the solutions, including \(L^\infty\)-bounds for the first derivatives by the maximum principle and estimates for the Hölder seminorms of the first derivatives by the Krylov-Safonov's Harnack inequality. As consequence, the Dirichlet problem for these equations is solved by the method of continuity.NEWLINENEWLINENEWLINEEquations of prescribed mean curvature are discussed in Chapter 3. Solvability of the Dirichlet problem is obtained as result of the global gradient estimates derived.NEWLINENEWLINENEWLINEChapter 4 is devoted to the minimal surface equations. Even if these are particular class of the mean curvature equations, the results in this chapter are proved by analysis ``upon surfaces''. The author treats minimal surfaces as submanifolds in the ambient Euclidean space and writes equations on these submanifolds.NEWLINENEWLINENEWLINEThe study of the fully nonlinear equations starts from Chapter 5 where various a priori estimates are derived for the solution and its first and second derivatives via the maximum principle and the result of Krylov and Safonov, and classical solvability of the Dirichlet problem is shown as well.NEWLINENEWLINENEWLINEChapter 6 deals with real Monge-Ampère equations. In particular, the global Hessian estimates obtained allow to apply the method of continuity in order to prove solvability of the Dirichlet problem. The structure of the equations is important here to overcome the lack of uniform ellipticity in this case.NEWLINENEWLINENEWLINEThese results are extended to complex Monge-Ampère equations in Chapter 7.NEWLINENEWLINENEWLINEGeneralized solutions of real Monge-Ampère equations are discussed in Chapter 8. Such solutions are defined only for convex functions which are not assumed to be \(C^2\) at the very beginning. The author proves various regularity results under appropriate assumptions on the corresponding Monge-Ampère measures. In particular, strict convexity and interior \(C^{1,\alpha}\)-regularity of the solution are proved if the Monge-Ampère measures satisfy the doubling condition. Optimal interior \(C^{2,\alpha}\)-regularity is proved via level set approach when the Monge-Ampère measures are induced by positive Hölder continuous functions.NEWLINENEWLINENEWLINESummarizing, the book of Han will serve as a valuable resource for graduate students and for anyone interested in the subject of nonlinear second order elliptic PDEs.