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Convex solutions of certain elliptic equations have constant rank Hessians. (English) Zbl 0624.35031
In recent years there have been several results on the convexity of solutions to certain elliptic equations, e.g. N. Korevaar [Indiana Univ. Math. J. 32, 603-614 (1983; Zbl 0481.35024)], A. Kennington [ibid. 34, 687-704 (1985; Zbl 0549.35025)] and the reviewer [Z. Angew. Math. Mech. 64, No.5, T 364-T 366 (1984; Zbl 0581.73006); Math. Methods Appl. Sci. 8, 93-101 (1986; Zbl 0616.35006); Commun. Partial Differ. Equations 10, 1213-1225 (1985; Zbl 0587.35026)]. In contrast to these results the authors of the paper under review assume that u is convex and solves \(\Delta u=f(u,\nabla u)>0\) in \(\Omega \subset {\mathbb{R}}^ n\). The main result is the following. If 1/f(\(\cdot,\nabla u)\) is convex in u, then the Hessian H of u has constant rank in \(\Omega\). This implies in particular, that H has full rank in \(\Omega\) if it has full rank in a single point of \(\Omega\). The proof is a nontrivial extension of a former result of L. Caffarelli and A. Friedman [Duke Math. J. 52, 431-456 (1985; Zbl 0599.35065)]. The paper contains also applications of the main result.
Reviewer: B.Kawohl

MSC:
35J60 Nonlinear elliptic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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