{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,9,8]],"date-time":"2024-09-08T11:43:40Z","timestamp":1725795820724},"publisher-location":"Dordrecht","reference-count":31,"publisher":"Springer Netherlands","isbn-type":[{"type":"print","value":"9789401790536"},{"type":"electronic","value":"9789401790543"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2014]]},"DOI":"10.1007\/978-94-017-9054-3_2","type":"book-chapter","created":{"date-parts":[[2014,6,19]],"date-time":"2014-06-19T07:01:47Z","timestamp":1403161307000},"page":"23-40","source":"Crossref","is-referenced-by-count":1,"title":["On the Numerical Solution of the Dirichlet Problem for the Elliptic $$(\\sigma _2)$$ ( \u03c3 2 ) Equation"],"prefix":"10.1007","author":[{"given":"Alexandre","family":"Caboussat","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2014,6,19]]},"reference":[{"issue":"68","key":"2_CR1","first-page":"89","volume":"2","author":"AD Aleksandrov","year":"1968","unstructured":"Aleksandrov AD (1968) Uniqueness conditions and estimates for the solution of the Dirichlet problem. Am Math Soc Trans 2(68):89\u2013119","journal-title":"Am Math Soc Trans"},{"issue":"4","key":"2_CR2","doi-asserted-by":"publisher","first-page":"737","DOI":"10.1051\/m2an\/2010017","volume":"44","author":"JD Benamou","year":"2010","unstructured":"Benamou JD, Froese BD, Oberman AM (2010) Two numerical methods for the elliptic Monge-Amp\u00e8re equation. M2AN. Math Model Numer Anal 44(4):737\u2013758","journal-title":"Math Model Numer Anal"},{"issue":"3","key":"2_CR3","doi-asserted-by":"publisher","first-page":"1212","DOI":"10.1137\/040621740","volume":"46","author":"K B\u00f6hmer","year":"2008","unstructured":"B\u00f6hmer K (2008) On finite element methods for fully nonlinear elliptic equations of second order. SIAM J Numer Anal 46(3):1212\u20131249","journal-title":"SIAM J Numer Anal"},{"issue":"276","key":"2_CR4","doi-asserted-by":"publisher","first-page":"1979","DOI":"10.1090\/S0025-5718-2011-02487-7","volume":"80","author":"SC Brenner","year":"2011","unstructured":"Brenner SC, Gudi T, Neilan M, Sung LY (2011) $${{C}}^{0}$$ C 0 penalty methods for the fully nonlinear Monge-Amp\u00e8re equation. Math Comp 80(276):1979\u20131995","journal-title":"Math Comp"},{"issue":"5","key":"2_CR5","doi-asserted-by":"publisher","first-page":"979","DOI":"10.1051\/m2an\/2011067","volume":"46","author":"SC Brenner","year":"2012","unstructured":"Brenner SC, Neilan M (2012) Finite element approximations of the three dimensional Monge-Amp\u00e8re equation. ESAIM Math Model Numer Anal 46(5):979\u20131001","journal-title":"ESAIM Math Model Numer Anal"},{"issue":"3","key":"2_CR6","doi-asserted-by":"publisher","first-page":"780","DOI":"10.1051\/cocv\/2012033","volume":"19","author":"A Caboussat","year":"2013","unstructured":"Caboussat A, Glowinski R, Sorensen DC (2013) A least-squares method for the numerical solution of the Dirichlet problem for the elliptic Monge-Amp\u00e8re equation in dimension two. ESAIM Control Optim Calc Var 19(3):780\u2013810. doi: 10.1051\/cocv\/2012033","journal-title":"ESAIM Control Optim Calc Var"},{"key":"2_CR7","volume-title":"Fully nonlinear elliptic equations","author":"LA Caffarelli","year":"1995","unstructured":"Caffarelli LA, Cabr\u00e9 X (1995) Fully nonlinear elliptic equations. American Mathematical Society, Providence"},{"key":"2_CR8","unstructured":"Chang SYA, Han ZC, Yang P (2009) On the prescribing $$\\sigma _2$$ \u03c3 2 curvature equation on $$\\mathbb{S}^4$$ S 4 . Preprint arXiv:0911.0375"},{"key":"2_CR9","doi-asserted-by":"crossref","unstructured":"Crandall MG, Ishii H, Lions PL (1992) User\u2019s guide to viscosity solutions of second order partial differential equations. Bull Amer Math Soc (N.S.) 27(1):1\u201367","DOI":"10.1090\/S0273-0979-1992-00266-5"},{"issue":"13\u201316","key":"2_CR10","doi-asserted-by":"publisher","first-page":"1344","DOI":"10.1016\/j.cma.2005.05.023","volume":"195","author":"EJ Dean","year":"2006","unstructured":"Dean EJ, Glowinski R (2006) Numerical methods for fully nonlinear elliptic equations of the Monge-Amp\u00e8re type. Comput Methods Appl Mech Eng 195(13\u201316):1344\u20131386","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2_CR11","unstructured":"Dean EJ, Glowinski R (2008) On the numerical solution of the elliptic Monge-Amp\u00e8re equation in dimension two: a least-squares approach. In: Glowinski R, Neittaanm\u00e4ki P (eds) Partial differential equations: modeling and numerical simulation. Computational in Methods Application Science, vol 16. Springer, Dordrecht, pp 43\u201363"},{"issue":"2","key":"2_CR12","doi-asserted-by":"publisher","first-page":"1226","DOI":"10.1137\/070710378","volume":"47","author":"X Feng","year":"2009","unstructured":"Feng X, Neilan M (2009) Mixed finite element methods for the fully nonlinear Monge-Amp\u00e8re equation based on the vanishing moment method. SIAM J Numer Anal 47(2):1226\u20131250","journal-title":"SIAM J Numer Anal"},{"issue":"3","key":"2_CR13","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/s10915-010-9439-1","volume":"47","author":"X Feng","year":"2011","unstructured":"Feng X, Neilan M (2011) Analysis of Galerkin methods for the fully nonlinear Monge-Amp\u00e8re equation. J Sci Comput 47(3):303\u2013327","journal-title":"J Sci Comput"},{"issue":"3","key":"2_CR14","doi-asserted-by":"publisher","first-page":"818","DOI":"10.1016\/j.jcp.2010.10.020","volume":"230","author":"BD Froese","year":"2011","unstructured":"Froese BD, Oberman AM (2011) Fast finite difference solvers for singular solutions of the elliptic Monge-Amp\u00e8re equation. J Comput Phys 230(3):818\u2013834","journal-title":"J Comput Phys"},{"key":"2_CR15","doi-asserted-by":"crossref","unstructured":"Gilbarg D, Trudinger NS (2001) Elliptic partial differential equations of second order. Springer, Berlin (Reprint of the 1998 edition)","DOI":"10.1007\/978-3-642-61798-0"},{"key":"2_CR16","first-page":"3","volume-title":"Handbook of numerical analysis","author":"R Glowinski","year":"2003","unstructured":"Glowinski R (2003) Finite element methods for incompressible viscous flow. In: Ciarlet PG, Lions JL (eds) Handbook of numerical analysis, vol IX. North-Holland, Amsterdam, pp 3\u20131176"},{"key":"2_CR17","doi-asserted-by":"crossref","unstructured":"Glowinski R (2008) Numerical methods for nonlinear variational problems. Springer, Berlin (Reprint of the 1984 original)","DOI":"10.1007\/978-3-662-12613-4"},{"key":"2_CR18","doi-asserted-by":"crossref","unstructured":"Glowinski R (2009) Numerical methods for fully nonlinear elliptic equations. In: ICIAM 07\u20136th international congress on industrial and applied mathematics, European Mathematical Society, Z\u00fcrich, pp 155\u2013192","DOI":"10.4171\/056-1\/9"},{"key":"2_CR19","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970838","volume-title":"Augmented Lagrangian and operator-splitting methods in nonlinear mechanics","author":"R Glowinski","year":"1989","unstructured":"Glowinski R, Le Tallec P (1989) Augmented Lagrangian and operator-splitting methods in nonlinear mechanics. SIAM, Philadelphia"},{"key":"2_CR20","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511721595","volume-title":"Exact and approximate controllability for distributed parameter systems: a numerical approach, Encyclopedia of Mathematics and its Applications","author":"R Glowinski","year":"2008","unstructured":"Glowinski R, Lions JL, He J (2008) Exact and approximate controllability for distributed parameter systems: a numerical approach, Encyclopedia of Mathematics and its Applications, vol 117. Cambridge University Press, Cambridge"},{"key":"2_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0195-3","volume-title":"The Monge-Amp\u00e8re equation","author":"CE Guti\u00e9rrez","year":"2001","unstructured":"Guti\u00e9rrez CE (2001) The Monge-Amp\u00e8re equation. Birkha\u00fcser, Boston, MA"},{"issue":"187","key":"2_CR22","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1090\/S0025-5718-1989-0969489-7","volume":"53","author":"WH Hundsdorfer","year":"1989","unstructured":"Hundsdorfer WH, Verwer JG (1989) Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems. Math Comp 53(187):81\u2013101","journal-title":"Math Comp"},{"issue":"9","key":"2_CR23","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1016\/j.crma.2007.03.015","volume":"344","author":"B Mohammadi","year":"2007","unstructured":"Mohammadi B (2007) Optimal transport, shape optimization and global minimization. C R Math Acad Sci Paris 344(9):591\u2013596","journal-title":"C R Math Acad Sci Paris"},{"issue":"3","key":"2_CR24","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1007\/s00211-009-0283-x","volume":"115","author":"M Neilan","year":"2010","unstructured":"Neilan M (2010) A nonconforming Morley finite element method for the fully nonlinear Monge-Amp\u00e8re equation. Numer Math 115(3):371\u2013394","journal-title":"Numer Math"},{"issue":"1","key":"2_CR25","doi-asserted-by":"publisher","first-page":"221","DOI":"10.3934\/dcdsb.2008.10.221","volume":"10","author":"AM Oberman","year":"2008","unstructured":"Oberman AM (2008) Wide stencil finite difference schemes for the elliptic Monge-Amp\u00e8re equation and functions of the eigenvalues of the Hessian. Discrete Contin Dyn Syst Ser B 10(1):221\u2013238","journal-title":"Discrete Contin Dyn Syst Ser B"},{"issue":"3","key":"2_CR26","doi-asserted-by":"publisher","first-page":"1058","DOI":"10.1137\/100798715","volume":"33","author":"M Picasso","year":"2011","unstructured":"Picasso M, Alauzet F, Borouchaki H, George PL (2011) A numerical study of some Hessian recovery techniques on isotropic and anisotropic meshes. SIAM J Sci Comput 33(3):1058\u20131076","journal-title":"SIAM J Sci Comput"},{"issue":"1","key":"2_CR27","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1007\/s11075-009-9300-5","volume":"53","author":"DC Sorensen","year":"2010","unstructured":"Sorensen DC, Glowinski R (2010) A quadratically constrained minimization problem arising from PDE of Monge-Amp\u00e8re type. Numer Algorithms 53(1):53\u201366","journal-title":"Numer Algorithms"},{"issue":"1","key":"2_CR28","first-page":"42","volume":"153","author":"AN Tikhonov","year":"1963","unstructured":"Tikhonov AN (1963) The regularization of incorrectly posed problems. Dokl Akad Nauk SSSR 153(1):42\u201352 (English transl.: Soviet Math Dokl 4(6):1624\u20131627, 1963)","journal-title":"Dokl Akad Nauk SSSR"},{"issue":"2","key":"2_CR29","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1007\/BF00375406","volume":"111","author":"NS Trudinger","year":"1990","unstructured":"Trudinger NS (1990) The Dirichlet problem for the prescribed curvature equations. Arch Ration Mech Anal 111(2):153\u2013179","journal-title":"Arch Ration Mech Anal"},{"key":"2_CR30","doi-asserted-by":"crossref","unstructured":"Trudinger NS (1997) Weak solutions of Hessian equations. Comm Partial Differ Equ 22 (7\u20138):1251\u20131261","DOI":"10.1080\/03605309708821299"},{"issue":"3","key":"2_CR31","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1002\/cpa.20251","volume":"62","author":"M Warren","year":"2009","unstructured":"Warren M, Yuan Y (2009) Hessian estimates for the sigma-2 equation in dimension 3. Comm Pure Appl Math 62(3):305\u2013321","journal-title":"Comm Pure Appl Math"}],"container-title":["Computational Methods in Applied Sciences","Modeling, Simulation and Optimization for Science and Technology"],"original-title":[],"link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-94-017-9054-3_2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,8,21]],"date-time":"2020-08-21T04:12:51Z","timestamp":1597983171000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-94-017-9054-3_2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"ISBN":["9789401790536","9789401790543"],"references-count":31,"URL":"https:\/\/doi.org\/10.1007\/978-94-017-9054-3_2","relation":{},"ISSN":["1871-3033"],"issn-type":[{"type":"print","value":"1871-3033"}],"subject":[],"published":{"date-parts":[[2014]]}}}