Partial regularity for symmetric quasiconvex functionals on BD

Journal de Mathématiques Pures et Appliquées - Tập 145 - Trang 83-129 - 2021
Franz Gmeineder1
1Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

Tài liệu tham khảo

Acerbi, 1987, A regularity theorem for minimizers of quasiconvex integrals, Arch. Ration. Mech. Anal., 99, 261, 10.1007/BF00284509 Acerbi, 2002, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164, 213, 10.1007/s00205-002-0208-7 Ambrosio, 1997, Fine properties of functions with bounded deformation, Arch. Ration. Mech. Anal., 139, 201, 10.1007/s002050050051 Alberti, 1993, Rank one property for derivatives of functions with bounded variation, Proc. R. Soc. Edinb., Sect. A, 123, 239, 10.1017/S030821050002566X Ambrosio, 1992, On the relaxation in BV(Ω;Rm) of quasi–convex integrals, J. Funct. Anal., 109, 76, 10.1016/0022-1236(92)90012-8 Ambrosio, 2000, Functions of Bounded Variation and Free Discontinuity Problems Anzellotti, 1980, Existence of the displacements field for an elastoplastic body subject to Hencky's law and Von Mises yield condition, Manuscr. Math., 32, 101, 10.1007/BF01298185 Anzellotti, 1988, Convex functionals and partial regularity, Arch. Ration. Mech. Anal., 102, 243, 10.1007/BF00281349 Arroyo-Rabasa Babadjian, 2015, Traces of functions of bounded deformation, Indiana Univ. Math. J., 64, 1271, 10.1512/iumj.2015.64.5601 Barroso, 2000, A relaxation theorem in the space of functions of bounded deformation, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), 29, 19 Beck, 2016, Elliptic Regularity - A First Course, vol. 19 Bildhauer, 2003, Convex Variational Problems – Linear, Nearly Linear and Anisotropic Growth Conditions, vol. 1818 Bourgain, 2004, H12 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation, Publ. Math. Inst. Hautes Études Sci., 99, 1, 10.1007/s10240-004-0019-5 Breit, 2012, Sharp conditions for korn inequalities in Orlicz spaces, J. Math. Fluid Mech., 14, 565, 10.1007/s00021-011-0082-x Breit, 2020, On the trace operator for functions of bounded A-variation, Anal. PDE, 13, 559, 10.2140/apde.2020.13.559 Buckley, 1994, Sobolev-Poincaré inequalities for p<1, Indiana Univ. Math. J., 43, 221, 10.1512/iumj.1994.43.43011 Carozza, 1998, Partial regularity of minimizers of quasiconvex integrals with subquadratic growth, Ann. Mat. Pura Appl. (4), 175, 141, 10.1007/BF01783679 Christiansen, 1979 Ciarlet, 1998, Inequalities of Korn's type on surfaces, vol. 140, 105 Ciarlet, 2006, A nonlinear Korn inequality on a surface, J. Math. Pures Appl., 85, 2, 10.1016/j.matpur.2005.10.010 Ciarlet, 2015, Nonlinear Korn inequalities, J. Math. Pures Appl., 104, 1119, 10.1016/j.matpur.2015.07.007 S. Conti, D. Faraco, F. Maggi, a new approach to counterexamples to L 1 estimates: Korn's inequality, geometric rigidity, and regularity for gradients of separately convex functions, Archive for rational mechanics and analysis 175 (2), 287–300. Conti, 2019, A note on the Hausdorff dimension of the singular set of solutions to elasticity type systems, Commun. Contemp. Math., 21, 10.1142/S0219199719500263 Dacorogna, 1982, Weak Continuity and Weak Lower Semicontinuity of Nonlinear Func- Tionals, vol. 922 Dal Maso, 2004, Higher order quasiconvexity reduces to quasiconvexity, Arch. Ration. Mech. Anal., 171, 55, 10.1007/s00205-003-0278-1 De Philippis, 2016, On the structure of A—free measures and applications, Ann. Math. (2), 184, 1017, 10.4007/annals.2016.184.3.10 De Philippis, 2017, Characterization of generalized Young measures generated by symmetric gradients, Arch. Ration. Mech. Anal., 224, 1087, 10.1007/s00205-017-1096-1 Diening, 2008, Fractional estimates for non-differentiable elliptic systems with general growth, Forum Math., 523 Diening, 2012, Partial regularity for minimizers of functionals with general growth, SIAM J. Math. Anal., 44, 3594, 10.1137/120870554 Duzaar, 2000, Optimal interior partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation, Manuscr. Math., 103, 267, 10.1007/s002290070007 Duzaar, 2002, Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals, J. Reine Angew. Math., 546, 73 Duzaar, 2005, Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth, Ann. Mat. Pura Appl. (4), 184, 421, 10.1007/s10231-004-0117-5 Ekeland, 1974, On the variational principle, J. Math. Anal. Appl., 47, 324, 10.1016/0022-247X(74)90025-0 Evans, 1986, Quasiconvexity and partial regularity in the calculus of variations, Arch. Ration. Mech. Anal., 95, 227, 10.1007/BF00251360 Fonseca, 1993, Relaxation of quasiconvex functionals in BV(Ω;Rp) for integrands f(x,u,∇u), Arch. Ration. Mech. Anal., 123, 1, 10.1007/BF00386367 Fonseca, 1999, A-quasiconvexity, lower semicontinuity, and Young measures, SIAM J. Math. Anal., 30, 1355, 10.1137/S0036141098339885 Friedrichs, 1947, On the boundary value problems of the theory of elasticity and Korn's inequality, Ann. Math., 48, 441, 10.2307/1969180 Fuchs, 2000, Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids, vol. 1749 Giaquinta, 1983, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, vol. 105 Giusti, 2003 Gmeineder, 2016, Symmetric-convex functionals of linear growth, J. Elliptic Parabolic Equ., 2, 59, 10.1007/BF03377392 Gmeineder, 2020, The regularity of minima for the Dirichlet problem on BD, Arch. Ration. Mech. Anal., 237, 1099, 10.1007/s00205-020-01507-5 Gmeineder, 2019, Sobolev regularity for convex functionals on BD, J. Calc. Var., 58, 56, 10.1007/s00526-019-1491-6 Gmeineder, 2019, Partial regularity for BV minimizers, Arch. Ration. Mech. Anal., 232, 1429, 10.1007/s00205-018-01346-5 Gmeineder, 2019, On critical Lp-differentiability of BD-maps, Rev. Mat. Iberoam., 35, 2071, 10.4171/rmi/1111 Gobert, 1962, Une inéquation fondamentale de la théorie de l'élasticité, Bull. Soc. R. Sci. Liège, 3–4, 182 Kirchheim, 2016, On rank one convex functions that are homogeneous of degree one, Arch. Ration. Mech. Anal., 221, 527, 10.1007/s00205-016-0967-1 Kohn, 1979 Kohn, 1982, New integral estimates for deformations in terms of their nonlinear strains, Arch. Ration. Mech. Anal., 78, 131, 10.1007/BF00250837 Kolyada, 1993, On the embedding of Sobolev spaces, Mat. Zametki, 54, 48 Kristensen, 2007, The singular set of Lipschitzian minima of multiple integrals, Arch. Ration. Mech. Anal., 184, 341, 10.1007/s00205-006-0036-2 Kristensen, 2010, Relaxation of signed integral functionals in BV, Calc. Var. Partial Differ. Equ., 37, 29, 10.1007/s00526-009-0250-5 Kuusi, 2016, Partial regularity and potentials, J. Éc. Polytech. Math., 3, 309, 10.5802/jep.35 Maz'ya, 2009, Theory of Sobolev Multipliers, vol. 307 Mingione, 2003, The singular set of solutions to non-differentiable elliptic systems, Arch. Ration. Mech. Anal., 166, 287, 10.1007/s00205-002-0231-8 Mingione, 2003, Bounds for the singular set of solutions to non linear elliptic systems, Calc. Var. Partial Differ. Equ., 18, 373, 10.1007/s00526-003-0209-x Mingione, 2006, Regularity of minima: an invitation to the dark side of the calculus of variations, Appl. Math., 51, 355, 10.1007/s10778-006-0110-3 Morrey, 2008, Multiple integrals in the calculus of variations Mosolov, 1972, On the correctness of boundary value problems in the mechanics of continuous media, Math. USSR Sb., 17, 257, 10.1070/SM1972v017n02ABEH001503 Nečas, 1966, Sur les normes équivalentes dans Wkp(Ω) et sur la coecivité des formes formellement positives, 102 Ornstein, 1962, A non-equality for differential operators in the L1-norm, Arch. Ration. Mech. Anal., 11, 40, 10.1007/BF00253928 Reshetnyak, 1968, Weak convergence of completely additive vector functions on a set, Sib. Mat. Zh., 9, 1386, 10.1007/BF02196453 Rindler, 2011, Lower semicontinuity for integral functionals in the space of functions of bounded deformation via rigidity and Young measures, Arch. Ration. Mech. Anal., 202, 63, 10.1007/s00205-011-0408-0 Schmidt, 2014, Partial regularity for degenerate variational problems and image restoration models in BV, Indiana Univ. Math. J., 63, 213, 10.1512/iumj.2014.63.5174 Schmidt, 2015 Schmidt, 2009, A simple partial regularity proof for minimizers of variational integrals, Nonlinear Differ. Equ. Appl., 16, 109, 10.1007/s00030-008-8012-1 Strang, 1981, Functions of bounded deformation, Arch. Ration. Mech. Anal., 75, 7 Strauss, 1973, Variations of Korn's and Sobolev's inequalities, vol. XXIII, 207 Suquet, 1979, Un espace fonctionnel pour les équations de la plasticité, Ann. Fac. Sci. Toulouse Math. (5), 1, 77, 10.5802/afst.531 Triebel, 1983 Van Schaftingen, 2013, Limiting Sobolev inequalities for vector fields and canceling linear differential operators, J. Eur. Math. Soc., 15, 877, 10.4171/JEMS/380 Zhang, 1992, A construction of quasiconvex functions with linear growth at infinity, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), 19, 313