推广散度定理的尖锐不等式--拟凸性概念的推广及补充
Sharp inequalities which generalize the divergence theorem--an extension of the notion of quasiconvexity, with an addendum
摘要：
在适当的边界条件下，得到了某些特殊的二次函数$f(\bf{E})$在区域$Ω$上的积分的尖锐不等式，其中$\bf{E}(\bf{x})$是由位势$\bf{U}(\bf{x})$导出的.利用$\bf{E}=\nabla\bf{U}$，已知当$f(\bf{E})$是拟凸函数并且当$\bf{U}$满足仿射边界条件（即，对于某些矩阵$\bf{D}$，$\partialΩ$上的$\bf{U}=\bf{D}\bf{x}$）。在这里，我们允许其他边界条件和领域$\bf{E}$，涉及衍生物的各种订单的$\bf{U}$。我们定义了一个推广拟凸性的凸性概念。介绍了$Q^*$ -凸二次函数的概念，给出了其特征，并给出了一个生成尖锐$Q^*$ -凸函数的算法.我们强调，这也解决了突出的问题，找到一个算法生成极值拟凸二次函数。我们还将积分$Ω$的特殊二次函数$g(\bf{J})$，其中$\bf{J}(\bf{x})$满足微分约束，涉及衍生物，可能，各种订单。本文的结果推广了Kang和作者在三维空间中的一个例子，其中$\bf{J}(\bf{x})$是满足$\nabla\cdot\bf{J}=0$的$3\times 3$矩阵值域.在增编中，对该文件进行了澄清。值得注意的是，更一般的边界条件下，尖锐的下界积分的某些二次函数的字段可以得到。因此，严格的$Q^*$ -凸性被认为是一个适当的条件，以确保广泛的一类常系数偏微分方程的解决方案的唯一性，在一个给定的域$Ω$适当的边界条件。
英文摘要：
Subject to suitable boundary conditions being imposed, sharp inequalities are obtained on integrals over a region $Ω$ of certain special quadratic functions $f(\bf{E})$ where $\bf{E}(\bf{x})$ derives from a potential $\bf{U}(\bf{x})$. With $\bf{E}=\nabla\bf{U}$ it is known that such sharp inequalities can be obtained when $f(\bf{E})$ is a quasiconvex function and when $\bf{U}$ satisfies affine boundary conditions (i.e., for some matrix $\bf{D}$, $\bf{U}=\bf{D}\bf{x}$ on $\partialΩ$). Here we allow for other boundary conditions and for fields $\bf{E}$ that involve derivatives of a variety orders of $\bf{U}$. We define a notion of convexity that generalizes quasiconvexity. $Q^*$-convex quadratic functions are introduced, characterized and an algorithm is given for generating sharply $Q^*$-convex functions. We emphasize that this also solves the outstanding problem of finding an algorithm for generating extremal quasiconvex quadratic functions. We also treat integrals over $Ω$ of special quadratic functions $g(\bf{J})$ where $\bf{J}(\bf{x})$ satisfies a differential constraint involving derivatives with, possibly, a variety of orders. The results generalize an example of Kang and the author in three spatial dimensions where $\bf{J}(\bf{x})$ is a $3\times 3$ matrix valued field satisfying $\nabla\cdot\bf{J}=0$. In the addendum the paper is clarified. Notably, much more general boundary conditions are given under which sharp lower bounds on the integrals of certain quadratic functions of the fields can be obtained. As a consequence, strict $Q^*$-convexity is found to be an appropriate condition to ensure uniqueness of the solutions of a wide class of partial differential equations with constant coefficients, in a given domain $Ω$ with appropriate boundary conditions.
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