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.

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) |