Dacorogna–Marcellini能量的拟凸性
Quasiconvexity for the Dacorogna--Marcellini Energy
浏览论文内容
中文总结 AI 辅助
该数学研究证明平面Dacorogna–Marcellini能量的拟凸性等价于秩-1凸性，条件为|γ|≤2/√3，其方法可推广到2×2矩阵上左右旋转不变的齐次四次多项式的相关性质。
中文摘要 AI 辅助
我们证明，平面Dacorogna–Marcellini能量$\\ f_\gamma(A)=|A|^4-2\gamma|A|^2\det A\\ $恰好是拟凸的，当且仅当它是秩-1凸的，即当且仅当$\\ |\gamma|\leq\frac{2}{\sqrt{3}}\\ $。证明使用了能量泛函沿分量热流的单调性性质。作为方法的推论，我们表明，对于2×2矩阵上左右旋转不变的齐次四次多项式，拟凸性等价于秩-1凸性。
英文摘要
We prove that the planar Dacorogna--Marcellini energy $f_γ(A)=|A|^4-2γ|A|^2\det A$ is quasiconvex exactly when it is rank-one convex, i.e. if and only if $|γ|\leq\frac{2}{\sqrt3}$. The proof uses a monotonicity property of the energy functional along the componentwise heat flow. As a corollary of our method, we show that for homogeneous quartic polynomials on $2 \times 2$ matrices invariant by left and right rotation, quasiconvexity is equivalent to rank one convexity.
