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    Zhang, Kewei 2011. On coercivity and regularity for linear elliptic systems. Calculus of Variations and Partial Differential Equations, Vol. 40, Issue. 1-2, p. 65.


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  • Bulletin of the Australian Mathematical Society, Volume 54, Issue 3
  • December 1996, pp. 423-430

On the coercivity of elliptic systems in two dimensional spaces

  • Kewei Zhang (a1) (a2)
  • DOI: http://dx.doi.org/10.1017/S0004972700021833
  • Published online: 01 April 2009
Abstract

We establish necessary conditions for quadratic forms corresponding to strongly elliptic systems in divergence form to have various coercivity properties in a smooth domain in ℝ2. We prove that if the quadratic form has some coercivity property, then certain types of BMO seminorms of the coefficients of the system cannot be very large. We use the connection between Jacobians and Hardy spaces and the special structures of elliptic quadratic forms defined on 2 X 2 matrices.

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[1]J.M. Ball , ‘Convexity conditions and existence theorems in nonlinear elasticity’, Arch. Rational Mech. Anal. 63 (1977), 337403.

[5]G. Geymonat , S. Müller and N. Triantafylldis , ‘Homognization of nonlinear elastic materials, microscopic bifurcation and macroscopic loss of rank-one convexity’, Arch. Rational Mech. Anal. 122 (1993), 231290.

[6]P. Jones , ‘Extension theorems for BMO’, Indiana Univ. Math. J. 29 (1980), 4166.

[7]P. Marcellini , ‘Quasiconvex quadratic forms in two dimensions’, Appl. Math. Optim. 11 (1984), 183189.

[8]D. Sarason , ‘Functions of vanishing mean oscillation’, Trans. Amer. Math. Soc. 207 (1975), 391405.

[9]F. Terpstra , ‘Die Darstellung biquadratischer formen als summen von quadraten mit anwendung auf die variations rechnung’, Math. Ann. 116 (1938), 166180.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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