Fully nonlinear second order elliptic equations: recent development. (English) Zbl 1033.35036
From the text: A short discussion of the history of the theory of fully nonlinear second-order elliptic equations is presented starting with the beginning of the century. Then an account of the explosion of results during the last decade is given. This explosion is based entirely on a generalization for nondivergence form linear operators of the celebrated De Giorgi result bearing an Hölder continuity. This is an extended version of a talk at Mathfest, Burlington, Vermont, Aug. 6–8, 1995.
The article is organized as follows. In Section 2, we give a general notion of fully nonlinear elliptic equations and some existence theorems. Section 3 is devoted to results bearing on the general theory of fully nonlinear uniformly elliptic equations and Section 4 contains a discussion of results for the general theory of fully nonlinear degenerate elliptic equations. In Section 5 we speak about equations related to the Monge-Ampère equations. Section 6 contains a new (and probably the first) result on the rate of convergence of numerical approximations for fully nonlinear degenerate elliptic equations.
We have already mentioned above that the modern theory of fully nonlinear elliptic equations is based entirely on some deep results from linear theory. Also many new brilliant ideas and techniques appeared. In Section 7 we present one of them which is Safonov’s proof of the Hölder-Korn-Lichtenstein-Giraud estimate for the Laplacian. This proof was designed for nonlinear equations and turned out to be shorter and easier than usual ones even for the simplest linear equation. In my opinion his proof should be part of general mathematical education.
Finally, not as brilliant and not a very popular technical idea is presented in Section 8. Exploiting this idea allowed the author to get some very general results on fully nonlinear degenerate elliptic equations. Also the idea is of a general character and might be of interest to mathematicians from other areas.
The article is organized as follows. In Section 2, we give a general notion of fully nonlinear elliptic equations and some existence theorems. Section 3 is devoted to results bearing on the general theory of fully nonlinear uniformly elliptic equations and Section 4 contains a discussion of results for the general theory of fully nonlinear degenerate elliptic equations. In Section 5 we speak about equations related to the Monge-Ampère equations. Section 6 contains a new (and probably the first) result on the rate of convergence of numerical approximations for fully nonlinear degenerate elliptic equations.
We have already mentioned above that the modern theory of fully nonlinear elliptic equations is based entirely on some deep results from linear theory. Also many new brilliant ideas and techniques appeared. In Section 7 we present one of them which is Safonov’s proof of the Hölder-Korn-Lichtenstein-Giraud estimate for the Laplacian. This proof was designed for nonlinear equations and turned out to be shorter and easier than usual ones even for the simplest linear equation. In my opinion his proof should be part of general mathematical education.
Finally, not as brilliant and not a very popular technical idea is presented in Section 8. Exploiting this idea allowed the author to get some very general results on fully nonlinear degenerate elliptic equations. Also the idea is of a general character and might be of interest to mathematicians from other areas.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |
References:
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