×

The Dirichlet problem for nonlinear second-order elliptic equations. II: Complex Monge-Ampère, and uniformly elliptic, equations. (English) Zbl 0598.35048

[For Part I, see ibid. 37, 369-402 (1984; Zbl 0598.35047).]
This is the second paper in a series of three papers devoted to the Dirichlet problem for second-order nonlinear elliptic equations. The third one will appear in Acta Mathematica. In this paper the authors treat the problem \[ F(x,u,Du,D^ 2u)=0\quad in\quad \Omega,\quad u=\phi \quad on\quad \partial \Omega. \] The function F is smooth for \(x\in {\bar \Omega}\) in all the arguments, \[ \sum (\partial F/\partial u_{ij})\xi_ i\xi_ j>0\quad for\quad \xi =(\xi_ 1,..,\xi_ n)\neq 0, \] and F is a concave function of the second derivatives \(\{u_{ij}\}\). (1) The paper contains three sections. In the first section the \(C^ 2\) a priori estimates for elliptic complex Monge- Ampère equations are derived. The principal contribution of the second section is the derivation of a logarithmic modulus of continuity of \(u_{ij}\) near the boundary. The last section is a self-contained treatment of a rather general class of ”uniformly elliptic” operators satisfying (1). It is worth mentioning that this paper is in close relation to works of N. V. Krylov and N. S. Trudinger which are stated in the references.
Reviewer: P.Drabek

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35B50 Maximum principles in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B45 A priori estimates in context of PDEs

Citations:

Zbl 0598.35047
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Springer, Berlin-Heidelberg- New York, 1982. · Zbl 0512.53044
[2] Beals, Bull. (new Ser.) Amer Math. Soc 8 pp 125– (1983)
[3] Bedford, Invent. Math. 50 pp 129– (1979)
[4] Bedford, Invent. Math 37 pp 1– (1976)
[5] Bedford, Duke Math. J. 45 pp 375– (1978)
[6] Amer. J. Math. 101 pp 1131– (1979)
[7] and , The Dirichlet problem for an equation of complex Monge-Ampère type, Partial Differential Equations and Geometry (ed. by ), Decker, 1979, pp. 39–50.
[8] Brezis, Arch. Rat’l. Mech. Anal. 71 pp 1– (1979)
[9] Caffarelli, Comm. Pure Appl. Math. 37 pp 369– (1984)
[10] Cheng, Comm. Pure Appl. Math. 33 pp 507– (1980)
[11] , and , Intrinsic norms on a complex manifold. Global Analysis, Papers in honor of Univ. of Tokyo Press, Tokyo, 1969, pp. 119–139.
[12] and , Degree theory on Banach manifolds, Nonlinear Functional Analysis. Proc. Symp. Pure Math. 18, Amer. Math. Soc., 1970, pp. 86–94.
[13] Evans, Comm. Pure Appl. Math. 35 pp 333– (1982)
[14] Evans, Trans. Amer. Math. Soc. 275 pp 245– (1983)
[15] Evans, Trans. Amer. Math. Soc. 253 pp 365– (1979)
[16] Evans, Ann. Inst. Fourier 31 pp 175– (1981) · Zbl 0441.35023
[17] and , Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin-Heidelberg-New York, 2d ed., 1984.
[18] Ivochkina, Ukrain. Math. J. 30 pp 32– (1978)
[19] Kerzman, Proc. Symp. Pure Math., Amer. Math. Soc. 30 pp 161– (1977)
[20] Krylov, Izvestia. Math. Ser. 46 pp 487– (1982)
[21] Engl. Trans. Math. USSR Izv. 20 pp 459– (1983)
[22] Krylov, Mat. Sbornik 120 pp 311– (1983)
[23] Engl. Trans. Math. USSR Sbornik 48 pp 307– (1984)
[24] Krylov, Izvestia Math. Ser. 47 pp 75– (1983)
[25] Krylov, Izvestia Akad. Nauk. SSSR 40 pp 161– (1980)
[26] Engl. Trans. in Math. USSR Izvestija 16 pp 151– (1981)
[27] and , Linear and quasilinear elliptic equations, Moscow: Izdat. ”nauka” 1964, 2nd ed. in 1973, Engl. Trans.: Academic Press, New York, 1968.
[28] Lempert, La metrique de Kobayashi et al representation des domaines sur la boule
[29] Lions, Acta Math. 146 pp 151– (1981)
[30] Lions, Proc. Int. Congress Math. Warsaw (1983)
[31] Lions, J. Control Optim. 20 pp 58– (1982)
[32] Optimal control of stochastic integrals and Hamilton-Jacobi-Bellman equations, II. SIAM p. 82–95. · Zbl 0478.93070
[33] Motzkin, J. Math. and Phys. 31 pp 253– (1952) · Zbl 0050.12501
[34] Pucci, Ann. Math. Pura Appl. 74 pp 141– (1966)
[35] Schulz, J. Reine Angew. Math.
[36] Smale, Amer. J. Math 87 pp 861– (1965)
[37] Trudinger, Invent. Math. 61 pp 67– (1980)
[38] Elliptic equations in non-Divergence form, Proc. Miniconference on Partial Differential Equations, Canberra 1, 1981, pp. 1–16.
[39] Trudinger, Trans. Amer Math. Soc.
[40] Yau, Comm. Pure Appl. Math. 31 pp 339– (1978)
[41] Krylov, Mat. Sbornik 121 pp 211– (1983)
[42] Engl. Trans. Math. USSR Sbornik 49 pp 207– (1984)
[43] Krylov, Dokl. Akad. Nauk 274 pp 23– (1984)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.