About Pogorelov's Method and Aleksandrov's Estimates
DOI:
https://doi.org/10.15407/mag16.03.283Ключові слова:
оцінка Александрова, метод Погорєлова, рівняння БеллманаАнотація
Ми даємо короткий огляд ролі оцінок Александрова та ідей Погорєлова у наших дослідженнях.Mathematics Subject Classification: 35-01, 35-03, 51-01
Посилання
A.D. Aleksandrov, Dirichlet’s problem for the equation det kzij k = ϕ(z1 , . . . , zn , z, x1 , . . . , xn ), I, Vestnik Leningrad. Univ., Ser. Mat., Meh., Astr., 13 (1958), No. 1, 5–24 (Russian).
I.Ja. Bakelman, Geometric Methods of Solution of Elliptic Equations, Nauka, Moscow, 1965 (Russian).
Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the Monge–Ampère equation det(∂ 2 u/∂xi ∂xj ) = F (x, u), Comm. Pure Appl. Math. 30 (1977), No. 1, 41–68.
N.V. Krylov, Sequences of convex functions and estimates of the maximum of the solution of a parabolic equation, Sibirsk. Mat. Zh., 17 (1976), No. 2, 290–303 (Russian); Engl. transl.: Sib. Math. J., 17 (1976), No. 2, 226–236. https://doi.org/10.1007/BF00967569
N.V. Krylov, Weighted Aleksandrov estimates: PDE and stochastic versions, Algebra i Analiz 31 (2019), No. 3, 154–169. https://doi.org/10.1090/spmj/1611
A.V. Pogorelov, The regularity of the generalized solutions of the equation det(∂ 2 u/∂xi ∂xj ) = ϕ(x1 , x2 , ..., xn ) > 0, Dokl. Akad. Nauk SSSR 200 (1971), 534– 537 (Russian); Engl. transl.: in Soviet Math. Dokl. 12 (1971), 1436–1440.
A.V. Pogorelov, The Dirichlet problem for the multidimensional analogue of the Monge–Ampère equation, Dokl. Akad. Nauk SSSR 201 (1971), 790–793 (Russian); Engl. transl.: Soviet Math. Dokl. 12 (1971), 1227–1231.
A.V. Pogorelov, The Minkowski Multidimensional Problem, Nauka, Moscow, 1975 (Russian); Engl. transl.: Scripta Series in Mathematics, V.H. Winston & Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-Toronto-London, 1978.
