An extension to a theorem of Jörgens, Calabi, and Pogorelov. (English) Zbl 1236.35041
Summary: We give some extensions to a theorem of K. Jörgens (\(n=2\)) [Math. Ann. 127, 130–134 (1954; Zbl 0055.08404)], E. Calabi (\(n\leq 5\)) [Mich. Math. J. 5, 105–126 (1958; Zbl 0113.30104)], and A. V. Pogorelov (\(n\geq 2\)) [Geom. Dedicata 1, 33–46 (1972; Zbl 0251.53005)]. The theorem asserts that any classical convex solution to \(\det(D^2u) = \) in \(\mathbb R^n\) must be a quadratic polynomial. One of our extensions asserts that for any positive Hölder continuous function \(f\) in \(\mathbb R^n\) which is 1-periodic in each variable, any convex solution of \(\det(D^2u) = f\) in \(\mathbb R^n\) must be the sum of a quadratic polynomial and a function which is 1-periodic in each variable.
MSC:
| 35J60 | Nonlinear elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
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