Delanoë, P. Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampère operator. (English) Zbl 0778.35037 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8, No. 5, 443-457 (1991). This work proves that two bounded strictly convex domains in \(\mathbb{R}^ 2\) can be mapped on one another by the gradient of a strictly convex function with Jacobian prescribed up to a constant, using the method of continuity. All data are smooth; the restriction \(n=2\) is used to get second-order boundary estimates.Note that a different approach to existence for this type of problem is via the factorization results of Y. Brenier [Commun. Pure Appl. Math. 44, No. 4, 375-417 (1991; Zbl 0738.46011)] as recently stressed by Caffarelli. Reviewer: S.Kichenassamy (Minneapolis) Cited in 3 ReviewsCited in 44 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35B45 A priori estimates in context of PDEs Keywords:Monge-Ampère equation; maps between convex domains Citations:Zbl 0738.46011 PDFBibTeX XMLCite \textit{P. Delanoë}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 8, No. 5, 443--457 (1991; Zbl 0778.35037) Full Text: DOI Numdam Numdam EuDML References: [1] Estimates Near the Boundary for Solutions of Elliptic Partial Differential Equations Satisfying General Boundary Conditions, II, Comm. Pure Appi. Math., Vol. 17, 35-92 (1964) · Zbl 0123.28706 [2] Aubin, T., Réduction du cas positif de l’équation de Monge-Ampère sur les variétés Kählériennes compactes à la démonstration d’une inégalité, J. Fune t. Anal., Vol. 53, 231-245 (1983) [3] Bakel’man, I., Generalized Solutions of Monge-Ampère Equations, Dokl. Akad. Nauk. S.S.S.R., Vol. 114, 6, 1143-1145 (1957), in russian · Zbl 0114.29602 [4] Caffarelli, L.; Nirenberg, L.; Spruck, J., The Dirichlet Problem for Nonlinear Second-Order Elliptic Equations I. Monge-Ampère equation, Comm. Pure Appi. Math., Vol. 37, 369-402 (1984) · Zbl 0598.35047 [5] Dacorogna, B.; Moser, J., On a Partial Differential Equation Involving the Jacobian Determinant, Ann. Inst. Henri Poincaré Analyse non linéaire, Vol. 7, 1, 1-26 (1990) · Zbl 0707.35041 [6] Delanoë, P., Equations du type de Monge-Ampère sur les variétés Riemanniennes compactes II, J. Funct. Anal., Vol. 41, 341-353 (1981) · Zbl 0474.58023 [7] Delanoë, P., Equations de Monge-Ampère en dimension deux, C. R. Acad. Sci. Paris, 294, série I, 693-696 (1982) · Zbl 0497.35039 [8] Delanoë, P., Plongements radiaux \(S^n\) → \(ℝ^{n + 1}\) à courbure de Gauss positive prescrite, Ann. Sci. Ec. Norm. Sup., Vol. 18, 635-649 (1985) · Zbl 0594.53039 [9] Delanoë, P., Remarques sur les variétés localement Hessiennes, Osaka J. Math., Vol. 26, 65-69 (1989) · Zbl 0754.53021 [10] Delanoë, P., Viscosity Solutions of Eikonal and Lie Equations on Compact Manifolds, Ann. Global Anal. Geom., Vol. 7, 2, 79-83 (1989) · Zbl 0644.58020 [11] Hopf, E., Elementare Bemerkungen über die Lösungen partieller Differential-gleichungen zweiter Ordnung vom elliptischen Typus, Sitz. Ber. Preu β. Akad. Wissensch. Berlin, Math.-Phys. KI, Vol. 19, 147-152 (1927) · JFM 53.0454.02 [12] Hopf, E., A Remark on Linear Elliptic Differential Equations of Second Order, Proc. Am. Math. Soc., Vol. 3, 791-793 (1952) · Zbl 0048.07802 [13] Ivotchkina, N. M., The a priori Estimate \(\| u \|_{C(\overline{\Omega})}^2\) on Convex Solutions of the Dirichlet problem for the Monge-Ampère Equation, Zapisk. Nautchn. Semin. LOMI, Vol. 96, 69-79 (1980) [14] Liao, L. Y.; Schulz, F., Regularity of Solutions of Two-Dimensional Monge-Ampère Equations, Transact. Am. Math. Soc., Vol. 307, 1, 271-277 (1988) · Zbl 0664.35023 [15] Lieberman, G. M.; Trudinger, N. S., Nonlinear Oblique Boundary Value Problems for Nonlinear Elliptic Equations, Transact. Am. Math. Soc., 295, 2, 509-546 (1986) · Zbl 0619.35047 [16] Lions, P.-L.; Trudinger, N. S.; Urbas, J. I.E., The Neumann problem for Equations of Monge-Ampère Type, Comm. Pure Appi. Math., Vol. 39, 539-563 (1986) · Zbl 0604.35027 [17] Nirenberg, L., On Nonlinear Elliptic Partial Differential Equations and Holder Continuity, Comm. Pure Appi. Math., Vol. 6, 103-156 (1953) · Zbl 0050.09801 [19] Schulz, F., Boundary Estimates for Solutions of Monge-Ampère Equations in the Plane, Ann. Sc. Norm. Sup. Pisa, Vol. 11, 3, 431-440 (1984) · Zbl 0573.35031 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.