Abstract

Using ideas from Compensated Compactness, we derive a necessary condition for any fourth degree polynomial on Rp\mathbb{R}^{p} to be sequentially lower semicontinuous with respect to weakly convergent fields defined on RN\mathbb{R}^N. We use that result to derive a necessary condition for the quasiconvexity of fourth degree polynomials of m×Nm\times N gradient matrices of vector fields defined on RN\mathbb{R}^N. This condition is violated by the example given by Šverák for m≥3m\geq 3 and N≥2N\geq 2, of a fourth degree polynomial which is rank-one convex, but it is not quasiconvex. These classes of functions are used in the approach to Nonlinear Elasticity based on the Calculus of Variations.

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Keywords

  • Compensated compactness
  • lower semicontinuity
  • quasiconvexity
  • rank-one convexity

Mathematics Subject Classification

15A15, 15A09, 15A23

References

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Suggested citation

S. Gutiérrez. “A Necessary Condition for the Quasiconvexity of Polynomials of Degree Four.” Journal of Convex Analysis 13 (2006), No. 1, 51–60. https://doi.org/10.68381/jca13004

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Published by Heldermann Verlag, 2006. Rights now held by Banach Press LLC.