Sharp inequalities which generalize the divergence theorem--an extension of the notion of quasiconvexity, with an addendum

Abstract

Subject to suitable boundary conditions being imposed, sharp inequalities are obtained on integrals over a region $\Omega$ 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 $\partial\Omega$). 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 $\Omega$ 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 $\Omega$ with appropriate boundary conditions.