On boundary Hölder gradient estimates for solutions to the linearized Monge-Ampère equations
HTML articles powered by AMS MathViewer
- by Nam Q. Le and Ovidiu Savin PDF
- Proc. Amer. Math. Soc. 143 (2015), 1605-1615 Request permission
Abstract:
In this paper, we establish boundary Hölder gradient estimates for solutions to the linearized Monge-Ampère equations with $L^{p}$ ($n<p\leq \infty$) right-hand side and $C^{1,\gamma }$ boundary values under natural assumptions on the domain, boundary data and the Monge-Ampère measure. These estimates extend our previous boundary regularity results for solutions to the linearized Monge-Ampère equations with bounded right-hand side and $C^{1, 1}$ boundary data.References
- Luis A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2) 130 (1989), no. 1, 189–213. MR 1005611, DOI 10.2307/1971480
- Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007, DOI 10.1090/coll/043
- Cristian E. Gutiérrez and Truyen Nguyen, Interior gradient estimates for solutions to the linearized Monge-Ampère equation, Adv. Math. 228 (2011), no. 4, 2034–2070. MR 2836113, DOI 10.1016/j.aim.2011.06.035
- Nam Q. Le and Truyen Nguyen, Global $W^{2,p}$ estimates for solutions to the linearized Monge-Ampère equations, Math. Ann. 358 (2014), no. 3-4, 629–700. MR 3175137, DOI 10.1007/s00208-013-0974-6
- N. Q. Le and O. Savin, Boundary regularity for solutions to the linearized Monge-Ampère equations, Arch. Ration. Mech. Anal. 210 (2013), no. 3, 813–836. MR 3116005, DOI 10.1007/s00205-013-0653-5
- Savin, O. A localization property at the boundary for the Monge-Ampère equation. Advances in Geometric Analysis, 45-68, Adv. Lect. Math. (ALM), 21, Int. Press, Somerville, MA, 2012.
- O. Savin, Pointwise $C^{2,\alpha }$ estimates at the boundary for the Monge-Ampère equation, J. Amer. Math. Soc. 26 (2013), no. 1, 63–99. MR 2983006, DOI 10.1090/S0894-0347-2012-00747-4
- O. Savin, Global $W^{2,p}$ estimates for the Monge-Ampère equation, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3573–3578. MR 3080179, DOI 10.1090/S0002-9939-2013-11748-X
- N. N. Ural′tseva, Hölder continuity of gradients of solutions of parabolic equations with boundary conditions of Signorini type, Dokl. Akad. Nauk SSSR 280 (1985), no. 3, 563–565 (Russian). MR 775926
- N. N. Ural′tseva, Estimates of derivatives of solutions of elliptic and parabolic inequalities, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 1143–1149. MR 934318
- Xu Jia Wang, Some counterexamples to the regularity of Monge-Ampère equations, Proc. Amer. Math. Soc. 123 (1995), no. 3, 841–845. MR 1223269, DOI 10.1090/S0002-9939-1995-1223269-0
- Xu-Jia Wang, Schauder estimates for elliptic and parabolic equations, Chinese Ann. Math. Ser. B 27 (2006), no. 6, 637–642. MR 2273802, DOI 10.1007/s11401-006-0142-3
Additional Information
- Nam Q. Le
- Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
- Address at time of publication: Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, 10307 Hanoi, Vietnam; Department of Mathematics, Indiana University, Bloomington, IN 47405
- MR Author ID: 839112
- Email: nqle@indiana.edu
- Ovidiu Savin
- Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
- MR Author ID: 675185
- Email: savin@math.columbia.edu
- Received by editor(s): May 17, 2013
- Received by editor(s) in revised form: August 8, 2013
- Published electronically: December 19, 2014
- Communicated by: Joachim Krieger
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 1605-1615
- MSC (2010): Primary 35J70, 35B65, 35B45, 35J96
- DOI: https://doi.org/10.1090/S0002-9939-2014-12340-9
- MathSciNet review: 3314073