Publication Cover
Applicable Analysis
An International Journal
Volume 99, 2020 - Issue 3
108
Views
6
CrossRef citations to date
0
Altmetric
Articles

Regularity for minimizers with positive Jacobian

, , &
Pages 496-507 | Received 08 Nov 2017, Accepted 20 Jul 2018, Published online: 06 Aug 2018

References

  • Ball JM. Convexity conditions and existence theorems in nonlinear elasticity. Arch Rational Mech Anal. 1976;63:337–403. doi: 10.1007/BF00279992
  • Dacorogna B. Direct methods in the calculus of variations. New York (NY): Springer; 2007. (Applied mathematical sciences; vol. 78).
  • Evans LC. Quasiconvexity and partial regularity in the calculus of variations. Arch Rational Mech Anal. 1986;95:227–252. doi: 10.1007/BF00251360
  • Acerbi E, Fusco N. A regularity theorem for minimizers of quasiconvex integrals. Arch Rational Mech Anal. 1987;99:261–281. doi: 10.1007/BF00284509
  • Giaquinta M, Modica G. Partial regularity of minimizers of quasiconvex integrals. Ann Inst Henri Poincaré, Analyse non lineaire. 1986;3:185–208. doi: 10.1016/S0294-1449(16)30385-7
  • Fusco N, Hutchinson J. Partial regularity in problems motivated by nonlinear elasticity. SIAM J Math Anal. 1991;22:1516–1551. doi: 10.1137/0522098
  • Fuchs M, Seregin G. Partial regularity of the deformation gradient for some model problems in nonlinear twodimensional elasticity. Algebra i Analiz. 1994;6:128–153.
  • Fuchs M, Reuling J. Partial regularity for certain classes of polyconvex functionals related to nonlinear elasticity. Manuscripta Math. 1995;87:13–26. doi: 10.1007/BF02570458
  • Passarelli di Napoli A. A regularity result for a class of polyconvex functionals. Ricerche Mat. 1999;48:379–393.
  • Esposito L, Mingione G. Partial regularity for minimizers of degenerate polyconvex energies. J Convex Anal. 2001;8:1–38.
  • Leonetti F. Maximum principle for vector-valued minimizers of some integral functionals. Boll Un Mat Ital. 1991;5-A:51–56.
  • Fusco N, Hutchinson JE. Partial regularity and everywhere continuity for a model problem from nonlinear elasticity. J Austral Math Soc (Ser A). 1994;57:158–169. doi: 10.1017/S1446788700037496
  • D'Ottavio A, Leonetti F, Musciano C. Maximum principle for vector valued mappings minimizing variational integrals. Atti Sem Math Univ Modena. 1998;46(Suppl.):677–683.
  • Leonetti F, Siepe F. Maximum principle for vector valued minimizers. J Convex Anal. 2005;12:267–278.
  • Leonetti F, Siepe F. Bounds for vector valued minimizers of some integral functionals. Ricerche Mat. 2005;54:303–312.
  • Cupini G, Leonetti F, Mascolo E. Local boundedness for minimizers of some polyconvex integrals. Arch Rational Mech Anal. 2017;224:269–289. doi: 10.1007/s00205-017-1074-7
  • Carozza M, Gao H, Giova R, et al., A boundedness result for minimizers of some polyconvex integrals. J Optim Theory Appl. 2018;178:699–725. DOI:10.1007/s10957-018-1335-0.
  • Bauman P, Phillips D. Univalent minimizers of polyconvex functionals in two dimensions. Arch Rational Mech Anal. 1994;126:161–181. doi: 10.1007/BF00391557
  • Leonetti F. Pointwise estimates for a model problem in nonlinear elasticity. Forum Math. 2006;18:529–534. doi: 10.1515/FORUM.2006.027
  • Ball JM, Murat F. W1,p quasiconvexity and variational problems for multiple integrals. J Funct Anal. 1984;58:225–253. doi: 10.1016/0022-1236(84)90041-7
  • Iwaniec T, Koskela P, Onninen J. Mapping of finite distortion: monotonicity and continuity. Invent Math. 2001;144:507–531. doi: 10.1007/s002220100130
  • Manfredi JJ. Weakly monotone functions. J Geom Anal. 1994;3:393–402. doi: 10.1007/BF02921588
  • Stein E, Weiss G. Introduction to Fourier analysis on Euclidean spaces. Princeton, NJ: Princeton University Press; 1971.
  • Bauman P, Owen NC, Phillips D. Maximum principles and a priori estimates for a class of problems from nonlinear elasticity. Ann Inst Henri Poincare Analyse Nonlineaire. 1991;8:119–157. doi: 10.1016/S0294-1449(16)30269-4
  • Fuchs M, Seregin G. Hölder continuity for weak estremals of two-dimensional variational problems related to nonlinear elasticity. Adv Math Sci Appl. 1997;7:413–425.
  • Bevan JJ. Explicit examples of Lipschitz, one-homogeneous solutions of log-singular planar elliptic systems. Nonlinear Anal. 2015;125:659–680. doi: 10.1016/j.na.2015.06.012
  • Bevan JJ. A condition for the Hölder regularity of local minimizers of a nonlinear elastic energy in two dimensions. Arch Rational Mech Anal. 2017;225:249–285. doi: 10.1007/s00205-017-1104-5
  • Bauman P, Phillips D, Owen NC. Maximal smoothness of solutions to certain Euler-Lagrange equations from nonlinear elasticity. Proc Roy Soc Edinb Sect A. 1991;119:241–263. doi: 10.1017/S0308210500014815
  • Bevan J, Yan X. Minimizers with topological singularities in two dimensional elasticity. ESAIM Control Optim Calc Var. 2008;14:192–209. doi: 10.1051/cocv:2007043
  • Yan X. Maximal smoothness for solutions to equilibrium equations in 2D nonlinear elasticity. Proc Amer Math Soc. 2007;135:1717–1724. doi: 10.1090/S0002-9939-06-08645-X
  • Stampacchia G. Equations elliptiques du second ordre a coefficientes discontinus. Semin. de Math. Superieures, Univ. de Montreal, Vol. 16, 1966.
  • Kovalevsky AA, Voitovich MV. On the improvement of summability of generalized solutions of the Dirichlet problem for nonlinear equations of the fourth order with strengthened ellipticity. Ukrainian Math J. 2006;58:1717–1733. doi: 10.1007/s11253-006-0164-8
  • Gao H, Leonetti F, Wang L. Remarks on Stampacchia Lemma. J Math Anal Appl. 2018;458:112–122. doi: 10.1016/j.jmaa.2017.08.056
  • Leonetti F, Petricca PV. Regularity for vector valued minimizers of some anisotropic integral functionals. JIPAM J Inequal Pure Appl Math. 2006;7(3). Article 88, 7 p.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.