  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.
