{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:15:22Z","timestamp":1758824122601,"version":"3.40.3"},"publisher-location":"Basel","reference-count":24,"publisher":"Springer Basel","isbn-type":[{"type":"print","value":"9783034801324"},{"type":"electronic","value":"9783034801331"}],"license":[{"start":{"date-parts":[[2011,10,28]],"date-time":"2011-10-28T00:00:00Z","timestamp":1319760000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2011,10,28]],"date-time":"2011-10-28T00:00:00Z","timestamp":1319760000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012]]},"DOI":"10.1007\/978-3-0348-0133-1_2","type":"book-chapter","created":{"date-parts":[[2012,1,2]],"date-time":"2012-01-02T10:34:06Z","timestamp":1325500446000},"page":"21-35","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Allen-Cahn and Cahn-Hilliard Variational Inequalities Solved with Optimization Techniques"],"prefix":"10.1007","author":[{"given":"Luise","family":"Blank","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Butz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Harald","family":"Garcke","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lavinia","family":"Sarbu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vanessa","family":"Styles","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2011,10,28]]},"reference":[{"key":"2_CR1","first-page":"290","volume":"213","author":"L Banas","year":"2009","unstructured":"L. Banas and R. N\u00fcrnberg, A multigrid method for the Cahn-Hilliard equation with obstacle potential. Appl. Math. Comput. 213 (2009), 290\u2013303.","journal-title":"Appl. Math. Comput."},{"issue":"1","key":"2_CR2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3934\/dcdss.2008.1.1","volume":"1","author":"JW Barrett","year":"2008","unstructured":"J.W. Barrett, H. Garcke, and R. N\u00fcrnberg, On sharp interface limits of Allen-Cahn\/Cahn-Hilliard variational inequalities. Discrete Contin. Dyn. Syst. S 1 (1) (2008), 1\u201314.","journal-title":"Discrete Contin. Dyn. Syst. S"},{"key":"2_CR3","unstructured":"L. Blank, H. Garcke, L. Sarbu, T. Srisupattarawanit, V. Styles, and A. Voigt, Phase-field approaches to structural topology optimization, a contribution in this book."},{"key":"2_CR4","doi-asserted-by":"crossref","unstructured":"L. Blank, M. Butz, and H. Garcke, Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method, ESAIM: Control, Optimization and Calculus of Variations, DOI: 10.1051\/COCV\/2010032.","DOI":"10.1051\/cocv\/2010032"},{"key":"2_CR5","unstructured":"L. Blank, H. Garcke, L. Sarbu, and V. Styles, Primal-dual active set methods for Allen-Cahn variational inequalities with non-local constraints. DFG priority program 1253 \u201cOptimization with PDEs\u201d, Preprint No. 1253-09-01."},{"key":"2_CR6","unstructured":"L. Blank, H. Garcke, L. Sarbu, and V. Styles, Non-local Allen-Cahn systems: Analysis and a primal dual active set method. Preprint No. 02\/2011, Universit\u00e4t Regensburg."},{"key":"2_CR7","unstructured":"L. Blank, L. Sarbu, and M. Stoll, Preconditioning for Allen-Cahn variational inequalities with non-local constraints. Preprint Nr.11\/2010, Universit\u00e4t Regensburg, Mathematik."},{"issue":"3","key":"2_CR8","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1017\/S095679250000053X","volume":"2","author":"JF Blowey","year":"1991","unstructured":"J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy I. Mathematical analysis. European J. Appl. Math. 2, no. 3 (1991), 233\u2013280.","journal-title":"European J. Appl. Math."},{"issue":"2","key":"2_CR9","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1017\/S0956792500000759","volume":"3","author":"JF Blowey","year":"1991","unstructured":"J.F. Blowey and C.M. Elliott, The Cahn-Hilliard gradient theory for phase separation with nonsmooth free energy II. Mathematical analysis. European J. Appl. Math. 3, no. 2 (1991), 147\u2013179.","journal-title":"European J. Appl. Math."},{"key":"2_CR10","first-page":"1","volume-title":"Motion by mean curvature and related topis (Trento, 1992)","author":"JF Blowey","year":"1994","unstructured":"J.F. Blowey and C.M. Elliott, A phase-field model with a double obstacle potential. Motion by mean curvature and related topis (Trento, 1992), 1\u201322, de Gruyter, Berlin 1994."},{"key":"2_CR11","doi-asserted-by":"crossref","first-page":"355","DOI":"10.1007\/BF00375607","volume":"124","author":"L Bronsard","year":"1993","unstructured":"L. Bronsard and F. Reitich, On three-phase boundary motion and the singular limit of a vector-valued Ginzburg-Landau equation. Arch. Rat. Mech. Anal. 124 (1993), 355\u2013379.","journal-title":"Arch. Rat. Mech. Anal."},{"key":"2_CR12","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1146\/annurev.matsci.32.112001.132041","volume":"32","author":"LQ Chen","year":"2001","unstructured":"L.Q. Chen, Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32 (2001), 113\u2013140.","journal-title":"Annu. Rev. Mater. Res."},{"issue":"8","key":"2_CR13","doi-asserted-by":"crossref","first-page":"1347","DOI":"10.1142\/S0218202508003066","volume":"18","author":"H Garcke","year":"2008","unstructured":"H. Garcke, B. Nestler, B. Stinner, and F. Wendler, Allen-Cahn systems with volume constraints. Mathematical Models and Methods in Applied Sciences 18, no. 8 (2008), 1347\u20131381.","journal-title":"Mathematical Models and Methods in Applied Sciences"},{"key":"2_CR14","doi-asserted-by":"crossref","first-page":"175","DOI":"10.4171\/IFB\/8","volume":"1","author":"H Garcke","year":"1999","unstructured":"H. Garcke, B. Nestler, and B. Stoth. Anisotropy in multi-phase systems: a phase field approach. Interfaces Free Bound. 1 (1999), 175\u2013198.","journal-title":"Interfaces Free Bound"},{"issue":"3","key":"2_CR15","doi-asserted-by":"crossref","first-page":"271","DOI":"10.4171\/IFB\/100","volume":"6","author":"H Garcke","year":"2004","unstructured":"H. Garcke and V. Styles. Bi-directional diffusion induced grain boundary motion with triple junctions. Interfaces Free Bound. 6, no. 3 (2004), 271\u2013294.","journal-title":"Interfaces Free Bound"},{"key":"2_CR16","first-page":"91","volume-title":"On preconditioned Uzawa-type iterations for a saddle point problem with inequality constraints","author":"C Gr\u00e4ser","year":"2007","unstructured":"C. Gr\u00e4ser and R. Kornhuber, On preconditioned Uzawa-type iterations for a saddle point problem with inequality constraints. Domain decomposition methods in science and engineering XVI, 91\u2013102, Lect. Notes Comput. Sci. Engl. 55, Springer, Berlin 2007."},{"issue":"3","key":"2_CR17","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1137\/S1052623401383558","volume":"13","author":"M Hinterm\u00fcller","year":"2002","unstructured":"M. Hinterm\u00fcller, K. Ito, and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim. 13 (2002), no. 3 (2003), 865\u2013888 (electronic).","journal-title":"SIAM J. Optim."},{"key":"2_CR18","unstructured":"M. Hinterm\u00fcller, M. Hinze, and M.H. Tber, An adaptive finite element Moreau-Yosida-based solver for a non-smooth Cahn-Hilliard problem. Matheon preprint Nr. 670, Berlin (2009), to appear in Optimization Methods and Software."},{"issue":"1","key":"2_CR19","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1051\/m2an:2003021","volume":"37","author":"K Ito","year":"2003","unstructured":"K. Ito and K. Kunisch, Semi-smooth Newton methods for variational inequalities of the first kind. M2AN Math. Model. Numer. Anal. 37, no. 1 (2003), 41\u201362.","journal-title":"M2AN Math. Model. Numer. Anal."},{"key":"2_CR20","unstructured":"R. Kornhuber and R. Krause. On multigrid methods for vector-valued Allen-Cahn equations with obstacle potential. Domain decomposition methods in science and engineering, 307\u2013314 (electronic), Natl. Auton. Univ. Mex., M\u00e9xico, 2003."},{"key":"2_CR21","doi-asserted-by":"crossref","first-page":"964","DOI":"10.1137\/0716071","volume":"16","author":"P-L Lions","year":"1979","unstructured":"P.-L. Lions and B. Mercier, Splitting algorithms for the sum of two non-linear operators. SIAM J. Numer. Anal. 16 (1979), 964\u2013979.","journal-title":"SIAM J. Numer. Anal."},{"key":"2_CR22","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/s10589-005-3056-1","volume":"33","author":"C Meyer","year":"2006","unstructured":"C. Meyer, A. R\u00f6sch, and F. Tr\u00f6ltzsch, Optimal control problems of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33, (2006), 209\u2013228.","journal-title":"Comput. Optim. Appl."},{"issue":"2","key":"2_CR23","first-page":"426","volume":"15","author":"I Neitzel","year":"2009","unstructured":"I. Neitzel and F. Tr\u00f6ltzsch, On regularization methods for the numerical solution of parabolic control problems with pointwise state constraints. ESAIM: COCV 15 (2) (2009), 426\u2013453.","journal-title":"ESAIM: COCV"},{"key":"2_CR24","first-page":"42","volume-title":"ALBERTA, Lecture Notes in Computational Science and Engineering","author":"A Schmidt","year":"2005","unstructured":"A. Schmidt and K.G. Siebert, Design of adaptive finite element software. The finite element toolbar. ALBERTA, Lecture Notes in Computational Science and Engineering 42, Springer-Verlag, Berlin 2005."}],"container-title":["International Series of Numerical Mathematics","Constrained Optimization and Optimal Control for Partial Differential Equations"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-0348-0133-1_2","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,8,13]],"date-time":"2020-08-13T07:20:57Z","timestamp":1597303257000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-0348-0133-1_2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10,28]]},"ISBN":["9783034801324","9783034801331"],"references-count":24,"URL":"https:\/\/doi.org\/10.1007\/978-3-0348-0133-1_2","relation":{},"ISSN":["0373-3149","2296-6072"],"issn-type":[{"type":"print","value":"0373-3149"},{"type":"electronic","value":"2296-6072"}],"subject":[],"published":{"date-parts":[[2011,10,28]]},"assertion":[{"value":"28 October 2011","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}}]}}