Regularity theory for quasilinear elliptic systems and Monge-Ampère equations in two dimensions
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.
scientific article; zbMATH DE number 48065
scientific article