Guan, Pengfei \(C^ 2\) a priori estimates for degenerate Monge-Ampère equations. (English) Zbl 0879.35059 Duke Math. J. 86, No. 2, 323-346 (1997). The author studies the degenerate case of the Dirichlet problem for the Monge-Ampère equation \[ \det(D^2u)= f\geq 0\quad\text{in }\Omega\subset\mathbb R^n,\quad u|_{\partial\Omega}=\phi. \] \(C^{1,1}\)-regularity and estimates are known in the case \(f\equiv 0\) [see N. S. Trudinger and J. Urbas, Bull. Aust. Math. Soc. 30, 321–334 (1984; Zbl 0557.35054)] or if \(f= g^n\) with some \(C^{1,1}\)-function \(g\) [see L. Caffarelli, J. J. Kohn, L. Nirenberg and J. Spruck, Commun. Pure Appl. Math. 38, 209–252 (1985; Zbl 0598.35048)].The author gives a new sufficient condition for \(f\), which allows him to prove \(C^{1,1}\)-estimates in the case \(\phi=0\). This condition is fulfilled e.g. for \(f=|g|\), \(g\in C^{1,1}\) if \(n=2\), or \(f\in C^{3,1}\) if \(n=3\). Also a nonvanishing boundary value is allowed, if. e.g. \(f\) is nondegenerate near the boundary. The technique is further extended to the equation of prescribed Gaussian curvature. Reviewer: M.Wiegner (Aachen) Cited in 1 ReviewCited in 36 Documents MSC: 35J70 Degenerate elliptic equations 35B45 A priori estimates in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35B65 Smoothness and regularity of solutions to PDEs Keywords:nonvanishing boundary value; equation of prescribed Gaussian curvature Citations:Zbl 0557.35054; Zbl 0598.35048 PDFBibTeX XMLCite \textit{P. Guan}, Duke Math. J. 86, No. 2, 323--346 (1997; Zbl 0879.35059) Full Text: DOI References: [1] L. Caffarelli, Some regularity properties of solutions of Monge-Ampère equation , Comm. Pure Appl. Math. 44 (1991), no. 8-9, 965-969. · Zbl 0761.35028 · doi:10.1002/cpa.3160440809 [2] L. Caffarelli, J. J. Kohn, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations , Comm. Pure Appl. Math. 38 (1985), no. 2, 209-252. · Zbl 0598.35048 · doi:10.1002/cpa.3160380206 [3] L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations, I: Monge-Ampère equation , Comm. Pure Appl. Math. 37 (1984), no. 3, 369-402. · Zbl 0598.35047 · doi:10.1002/cpa.3160370306 [4] L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for the degenerate Monge-Ampère equation , Rev. Mat. Iberoamericana 2 (1986), no. 1-2, 19-27. · Zbl 0611.35029 · doi:10.4171/RMI/23 [5] C. Fefferman and D. Phong, The uncertainty principle and sharp Garding inequalities , Comm. Pure Appl. Math. 34 (1981), no. 3, 285-331. · Zbl 0458.35099 · doi:10.1002/cpa.3160340302 [6] P. Guan, Regularity of a class of quasilinear degenerate elliptic equations , to appear in Adv. Math. · Zbl 0892.35036 · doi:10.1006/aima.1997.1677 [7] P. Guan and Y. Y. Li, The Weyl problem with nonnegative Gauss curvature , J. Differential Geom. 39 (1994), no. 2, 331-342. · Zbl 0796.53056 [8] P. Guan and Y. Y. Li, \(C^1,1\) estimates for solutions of a problem of Alexanderov , preprint, 1995. · Zbl 0879.53047 · doi:10.1002/(SICI)1097-0312(199708)50:8<789::AID-CPA4>3.0.CO;2-2 [9] N. Ivochkina, Solution of the Dirichlet problem for equations of \(m\) , Math. USSR Sb. 67 (1990), 317-339. · Zbl 0709.35046 · doi:10.1070/SM1990v067n02ABEH002089 [10] N. V. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain , Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 1, 75-108. · Zbl 0578.35024 · doi:10.1070/IM1984v022n01ABEH001434 [11] N. V. Krylov, Smoothness of the payoff function for a controllable diffusion process in a domain , Math. USSR Izv. 34 (1990), 65-95. · Zbl 0701.93054 · doi:10.1070/IM1990v034n01ABEH000603 [12] N. S. Trudinger and J. Urbas, The Dirichlet problem for the equation of prescribed Gauss curvature , Bull. Austral. Math. Soc. 28 (1983), no. 2, 217-231. · Zbl 0524.35047 · doi:10.1017/S000497270002089X [13] N. S. Trudinger and J. Urbas, On second derivative estimates for equations of Monge-Ampère type , Bull. Austral. Math. Soc. 30 (1984), no. 3, 321-334. · Zbl 0557.35054 · doi:10.1017/S0004972700002069 [14] X. Wang, Some counterexamples to the regularity of Monge-Ampère equations , Proc. Amer. Math. Soc. 123 (1995), no. 3, 841-845. JSTOR: · Zbl 0822.35054 · doi:10.2307/2160809 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.