부산대학교 도서관

When dealing with vectorial minimizers of functionals in the calculusof variations, a key concept for the semicontinuity is the notion ofquasiconvexity, introduced by Morrey in 1952. However, the drawback ofthis definition lies in the difficulty to check it even for simplerLagrangians. In some situations, such as for quadratic forms,quasiconvexity turns out to be equivalent to rank-one convexity. Stillanother condition, intermediate between convexity and quasiconvexity,is polyconvexity, introduced by J. Ball.\par The purpose of this article is to continue the analysis of quadraticforms using the link between different notions in the calculus ofvariations such as quasiconvexity, polyconvexity, rank-one convexityand corresponding interpretations in algebraic geometry. Quasiconvexquadratic forms in applied mathematics correspond to nonnegativebiquadratic forms in algebraic geometry, while polyconvex ones are sumsof squares in algebraic geometry. Let $\Cal{C}_{N, n}$ denote theconvex cone of $N\times n$ quasiconvex quadratic forms.\par An important concept was introduced by Milton and explored in a seriesof papers; see [20,21,22,23].\par Definition 1. A quasiconvex quadratic form $f(\xi)\: \Bbb{R}^{N\timesn}\to \Bbb{R}$ is called\roster\item"$\bullet$" a weak (or Milton) extremal if one can not subtract a convexform fromit, preserving the quasiconvexity;\item"$\bullet$" an extreme ray of $\Cal{C}_{N,n}$ (or a strong extremal) if onecannot subtract a quasiconvex form from it, preserving the quasiconvexityof $f$.\endroster\par Extremal quasiconvex forms play an important role in applications suchas bounds for effective tensors of electrical and magnetic propertiesof two-component two-dimensional composites. Another related problemconcerns quadratic forms for which the associated tensor (acoustictensor) has particular symmetry properties. In the case $n=N=3$, let $f$ be a quadratic form on $\Bbb R^{3\times 3}$ corresponding to alinear elastic cubic tensor $T$ of rank four, $ f(\xi)=\xi T \xi^t$where $\xi=\{\xi_{i,j}\}_{i,j=1}^3$. The first author and Milton showedthat if $f$ is quasiconvex, then it can be represented as a sum ofconvex and null-Lagrangian form. Next, the authors turn to quadraticforms with cyclic and axis-reflexion symmetry. A particular such formis$$q(\xi)=(\xi_{11}^2+\xi_{22}^2+\xi_{33}^2)-2(\xi_{11}\xi_{22}+\xi_{22}\xi_{33}+\xi_{33}\xi_{11})+(\xi_{12}^2+\xi_{23}^2+\xi_{31}^2).$$They establish that this quadratic form is quasiconvex but notpolyconvex; moreover, $q$ is shown to be strong extremal. In algebraicgeometry, the problem of expressing a nonnegative homogeneouspolynomial as a sum of squares was known as one of Hilbert's problemsand was solved earlier by Choi and Lam in [17].\par The first author and Milton found an interesting connection betweenextremal polynomials and extremal elasticity tensors that are at theboundary of being rank-one convex. They proved that an elasticitytensor with orthotropic symmetry is extreme if the determinant of itsacoustic tensor is an extremal polynomial that is not a perfect square.In addition, they have some results concerning weak extremals. Namely,in the case $d=3$ they proved that if the determinant of the acoustictensor of the form is identically zero, then the form is either weakextremal or polyconvex. Also, if the determinant of the acoustic tensorof the form is a perfect square, then the form is either weak extremalor polyconvex, or is a sum of a rank-1 form and an extremal whoseacoustic tensor determinant is identically zero.\par In the present paper the authors are able to upgrade some of theprevious results to the case of extreme rays (or strong extremal):\par Theorem 2. Let $f(\xi)= \xi C\xi^T \in \Cal{C}_3$, where $\xi \in\Bbb{R}^{3\times 3}$ and $C$ is a fourth-order tensor with the usualsymmetries: $ C_{ijkl}=C_{kjil}=C_{ilkj}$. Assume that the determinantof the $y$-matrix of $f(x\otimes y)$ is an extremal polynomial. Thenone has one of the following:\roster\item"$\bullet$" If the determinant of the acoustic tensor is not a perfectsquare,then $f$ must be an extreme ray of $\Cal{C}_3$.\item"$\bullet$" If the determinant of the acoustic tensor is a perfect square,then$f$ is either an extreme ray of $\Cal{C}_3$ or polyconvex.\endroster\par The main tool is the following result:\par Theorem 3. Let $f(\xi)= \xi C\xi^T \in \Cal{C}_3$, where $\xi \in\Bbb{R}^{3\times 3}$ and $C$ is a fourth-order tensor with the usualsymmetries. Assume that the determinant of the $y$-matrix of$f(x\otimes y)$ is identically zero. Then $f$ must be a polyconvexform.\par Next, based on some recent results of Helton [24], the authors conjecturethat any non-polyconvex weak extremal $f\in \Cal{C}_3$ is an extremeray of $\Cal{C}_3$, and if $f\in \Cal{C}_3$ is a non-polyconvex extremeray, then the determinant of the acoustic tensor is an extremalpolynomial different from a perfect square.