A Necessary Condition for the Quasiconvexity of Polynomials of Degree Four

Abstract

<jats:p> Using ideas from Compensated Compactness, we derive a necessary condition for any fourth degree polynomial on <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{R}^{p}</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> to be sequentially lower semicontinuous with respect to weakly convergent fields defined on <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{R}^N</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . We use that result to derive a necessary condition for the quasiconvexity of fourth degree polynomials of <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">m\times N</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>×</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> gradient matrices of vector fields defined on <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">\mathbb{R}^N</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> . This condition is violated by the example given by Šverák for <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">m\geq 3</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:tex-math notation="LaTeX">N\geq 2</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> </jats:alternatives> </jats:inline-formula> , 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. </jats:p>