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Regularity theory for quasilinear elliptic systems and Monge-Ampère equations in two dimensions. (English) Zbl 0709.35038
Lecture Notes in Mathematics, 1445. Berlin etc.: Springer-Verlag. xv, 123 p. DM 25.00 (1990).
These notes present the “characteristic theory” for elliptic Monge- Ampère equations $(*)\quad Ar+2Bs+Ct+(rt-s^ 2)=E,$ where z: $$\Omega\to {\mathbb{R}}$$, $$\Omega \subset {\mathbb{R}}^ 2$$, $$p=z_ x$$, $$q=z_ y$$, $$r=z_{xx}$$, $$s=z_{xy}$$, $$t=z_{yy}$$, A,...,E may depend on x, y, z, p, q.
This theory was largely developed by H. Lewy and E. Heinz in a number of original papers (too much to be cited here) and is presented in book form for the first time. The author includes some of his own contributions [see e.g.: Arch. Ration. Mech. Anal. 89, 123-133 (1985; Zbl 0572.35039)] and geometric applications [Regularity of locally convex surfaces, Bull. Aust. Math. Soc. (to appear)].
The main result is to be found in chapter 9 and reads as follows: Let $$z\in C^{1,1}(\Omega)$$ satisfy the uniformly elliptic Monge-Ampère equation (*) and assume some structure conditions and bounds for A,...,E.
Then $$z\in C^{2,\mu}_{loc}(\Omega)$$, and its $$C^{2,\mu}$$-norm can locally be estimated by bounds for its gradient.
The central idea to prove this is to introduce “characteristic parameters” u, v which uniformize the characteristic differential form (“Riemannian metric”) corresponding to (*) and whose inverse x(u,v), y(u,v) satisfies a quasilinear “Heinz-Lewy system”. Via the corresponding conformality relations, the desired result can be obtained by deriving gradient bounds and estimates for the Jacobian from below for solutions of Heinz-Lewy systems.
An example confirms, that the obtained results are sharp.
The content of the further chapters is as follows:
In chapters 1, 2 the needed results from the regularity theory for quasilinear elliptic systems (Campanato technique) are presented.
In chapter 3 the regularity result $$C^{1,1}\to C^{2,\mu}$$ is obtained quite directly by a “Legendre-like transformation”, which is based on some of the author’s work. But the corresponding local $$C^{2,\alpha}$$- estimate depends also on bounds for $$D^ 2z.$$
Chapter 4: Similarity principle and Harnack-type inequalities for pseudoanalytic functions.
In chapter 5 it is shown how to estimate the Jacobian form below for solutions of special Heinz-Lewy systems using the function theoretic results of chapter 4.
The general Heinz-Lewy systems are treated in chapter 8, using results of Hartman-Wintner and Heinz on the local behavior (asymptotic expansions near zeroes) of solutions to certain differential inequalities (chapter 7).
In chapter 6 a global uniformization theorem is presented, which links Monge-Ampère equations and Heinz-Lewy systems.
In the last chapter 10 some applications to differential geometry are given:
i) regularity and a-priori estimates for convex (concave) solutions of the prescribed Gauß curvature equation,
ii) regularity and a-priori estimates for locally convex surfaces, using conjugate isothermal parameters and the Darboux system.
These lecture notes grew out on a seminar and give a clearly written, largely selfcontained introduction to this interesting topic. They will surely motivate a lot of mathematicians to get interested in Monge- Ampère equations.
Reviewer: H.-Ch.Grunau

##### MSC:
 35J60 Nonlinear elliptic equations 35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C20 Global Riemannian geometry, including pinching 35D10 Regularity of generalized solutions of PDE (MSC2000) 35J45 Systems of elliptic equations, general (MSC2000)
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