We consider higher order functionals of the form where the integrand , m ≥ 1 is strictly quasiconvex and satisfies a non-standard growth condition. More precisely we assume that f fulfills the (p, q)-growth condition
Mots clés : higher order functionals, non-standard growth, regularity theory
@article{COCV_2011__17_2_472_0, author = {Schemm, Sabine}, title = {Partial regularity of minimizers of higher order integrals with $(p, q)$-growth}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {472--492}, publisher = {EDP-Sciences}, volume = {17}, number = {2}, year = {2011}, doi = {10.1051/cocv/2010016}, zbl = {1248.49053}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2010016/} }
TY - JOUR AU - Schemm, Sabine TI - Partial regularity of minimizers of higher order integrals with $(p, q)$-growth JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 472 EP - 492 VL - 17 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010016/ DO - 10.1051/cocv/2010016 LA - en ID - COCV_2011__17_2_472_0 ER -
%0 Journal Article %A Schemm, Sabine %T Partial regularity of minimizers of higher order integrals with $(p, q)$-growth %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 472-492 %V 17 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010016/ %R 10.1051/cocv/2010016 %G en %F COCV_2011__17_2_472_0
Schemm, Sabine. Partial regularity of minimizers of higher order integrals with $(p, q)$-growth. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 2, pp. 472-492. doi : 10.1051/cocv/2010016. http://www.numdam.org/articles/10.1051/cocv/2010016/
[1] Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86 (1984) 125-145. | MR | Zbl
and ,[2] A regularity theorem for minimizers of quasiconvex integrals. Arch. Ration. Mech. Anal. 99 (1987) 261-281. | MR | Zbl
and ,[3] Regularity for minimizers of non-quadratic functionals: the case 1<p<2. J. Math. Anal. Appl. 140 (1989) 115-135. | MR | Zbl
and ,[4] Partial regularity under anisotropic (p, q) growth conditions. J. Differ. Equ. 107 (1994) 46-67. | MR | Zbl
and ,[5] Regularity results for a class of quasiconvex functionals with nonstandard growth. Ann. Scuola Norm. Sup. Pisa Cl. Sci. IV 30 (2001) 311-339. | Numdam | MR | Zbl
and ,[6] W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | MR | Zbl
and ,[7] Partial regularity for variational integrals with (s, µ, q)-growth. Calc. Var. Partial Differ. Equ. 13 (2001) 537-560. | MR | Zbl
and ,[8] C1, α-solutions to non-autonomous anisotropic variational problems. Calc. Var. Partial Differ. Equ. 24 (2005) 309-340. | MR | Zbl
and ,[9] The effective bulk energy of the relaxed energy of multiple integrals below the growth exponent. Proc. Roy. Soc. Edinburgh Sect. A 128 (1998) 463-479. | MR | Zbl
, and ,[10] Partial regularity for anisotropic functionals of higher order. ESAIM: COCV 13 (2007) 692-706. | Numdam | MR | Zbl
and ,[11] Partial regularity of minimizers of quasiconvex integrals with subquadratic growth. Ann. Mat. Pura Appl. IV 175 (1998) 141-164. | MR | Zbl
, and ,[12] Regularity of minimizers of vectorial integrals with p-q growth. Nonlinear Anal., Theory Methods Appl. 54 (2003) 591-616. | MR | Zbl
, and ,[13] Regularity of ω-minimizers of quasi-convex variational integrals with polynomial growth. Differ. Geom. Appl. 17 (2002) 139-152. | MR | Zbl
and ,[14] Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals. J. Reine Angew. Math. 546 (2002) 73-138. | MR | Zbl
and ,[15] Partial regularity for almost minimizers of quasi-convex integrals. SIAM J. Math. Anal. 32 (2000) 665-687. | MR | Zbl
, and ,[16] Regularity of almost minimizers of quasi-convex variational integrals with subquadratic growth. Ann. Mat. Pura Appl. IV 184 (2005) 421-448. | MR | Zbl
, and ,[17] Relaxation results for higher order integrals below the natural growth exponent. Differ. Integral Equ. 15 (2002) 671-696. | MR | Zbl
and ,[18] Regularity results for minimizers of irregular integrals with (p, q) growth. Forum Math. 14 (2002) 245-272. | MR | Zbl
, and ,[19] Sharp regularity for functionals with (p, q) growth. J. Differ. Equ. 204 (2004) 5-55. | MR | Zbl
, and ,[20] Quasiconvexity and partial regularity in the calculus of variations. Arch. Ration. Mech. Anal. 95 (1986) 227-252. | MR | Zbl
,[21] Relaxation of multiple integrals below the growth exponent. Ann. Inst. Henri Poincaré Anal. Non Linéaire 14 (1997) 309-338. | Numdam | MR | Zbl
and ,[22] From jacobian to hessian: distributional form and relaxation. Riv. Mat. Univ. Parma 4 (2005) 45-74. | MR | Zbl
and ,[23] C1, α partial regularity of functions minimising quasiconvex integrals. Manuscr. Math. 54 (1984) 121-143. | MR | Zbl
and ,[24] Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton University Press, Princeton (1983). | MR | Zbl
,[25] Growth conditions and regularity, a counterexample. Manuscr. Math. 59 (1987) 245-248. | MR | Zbl
,[26] Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. Henri Poincaré Anal. Non Linéaire 3 (1986) 185-208. | Numdam | MR | Zbl
and ,[27] A remark on partial regularity of minimizers of quasiconvex integrals of higher order. Rend. Ist. Mat. Univ. Trieste 32 (2000) 1-24. | MR | Zbl
,[28] Lower semicontinuity of quasiconvex integrals of higher order. NoDEA 6 (1999) 227-246. | MR | Zbl
and ,[29] Some remarks on the minimizers of variational integrals with non standard growth conditions. Boll. Un. Mat. Ital. A 6 (1992) 91-101. | MR | Zbl
,[30] Lower semicontinuity in Sobolev spaces below the growth exponent of the integrand. Proc. Roy. Soc. Edinburgh Sect. A 127 (1997) 797-817. | MR | Zbl
,[31] The singular set of lipschitzian minima of multiple integrals. Arch. Ration. Mech. Anal. 184 (2007) 341-369. | MR | Zbl
and ,[32] Partial regularity results for minimizers of quasiconvex functionals of higher order. Ann. Inst. Henri Poincaré Anal. Non Linéaire 19 (2002) 81-112. | Numdam | MR | Zbl
,[33] Approximation of quasiconvex functions and lower semicontinuity of multiple integrals. Manuscr. Math. 51 (1985) 1-28. | MR | Zbl
,[34] On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. Henri Poincaré Anal. Non Linéaire 3 (1986) 391-409. | Numdam | MR | Zbl
,[35] Un exemple de solution discontinue d'un problème variationnel dans le cas scalaire. Preprint Istituto Matematico U. Dini, Universita' di Firenze (1987/1988), n. 11.
,[36] Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions. Arch. Ration. Mech. Anal. 105 (1989) 267-284. | MR | Zbl
,[37] Regularity and existence of solutions of elliptic equations with p, q-growth conditions. J. Differ. Equ. 90 (1991) 1-30. | MR | Zbl
,[38] Everywhere regularity for a class of elliptic systems without growth conditions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. IV 23 (1996) 1-25. | Numdam | MR | Zbl
,[39] Quasi-convexity and lower semi-continuity of multiple variational integrals of any order. Trans. Am. Math. Soc. 119 (1965) 125-149. | MR | Zbl
,[40] Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2 (1952) 25-53. | MR | Zbl
,[41] A regularity result for a class of anisotropic systems. Rend. Ist. Mat. Univ. Trieste 28 (1996) 13-31. | MR | Zbl
and ,[42] Partial regularity of strong local minimizers of quasiconvex integrals with (p, q)-growth. Proc. Roy. Soc. Edinburgh Sect. A 139 (2009) 595-621. | MR | Zbl
and ,[43] Regularity of minimizers of W1,p-quasiconvex variational integrals with (p, q)-growth. Calc. Var. Partial Differ. Equ. 32 (2008) 1-24. | MR | Zbl
,[44] Regularity of relaxed minimizers of quasiconvex variational integrals with (p, q)-growth. Arch. Ration. Mech. Anal. 193 (2009) 311-337. | MR | Zbl
,[45] Partial regularity for quasiconvex integrals of any order. Ric. Mat. 52 (2003) 31-54. | MR | Zbl
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