Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp inequalities which generalize the divergence theorem--an extension
  of the notion of quasiconvexity, with an addendum   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.

5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
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The categories of this paper are: chao-dyn
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The categories of this paper are: cmp-lg
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The categories of this paper are: physics
27
The categories of this paper are: comp-gas