1302.0942 (Graeme W. Milton)

Sharp inequalities which generalize the divergence theorem    [PDF]

Graeme W. Milton

Subject to suitable boundary conditions being imposed, sharp inequalities are obtained on integrals over a region \Omega of certain special quadratic functions f(E) where E(x) derives from a potential U(x). With E=Grad(U) it is known that such sharp inequalities can be obtained when f(E) is a quasiconvex function and when U satisfies affine boundary conditions (i.e., for some matrix D, U=Dx on the boundary of \Omega). Here we allow for other boundary conditions and for fields E that involve derivatives of a variety orders of U. We also treat integrals over \Omega of special quadratic functions g(J) where J(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 J(x) is a 3 by 3 matrix valued field satisfying Div(J)=0. We also present an algorithm for generating extremal quasiconvex quadratic functions.

View original: http://arxiv.org/abs/1302.0942