Sharp inequalities that generalize the divergence theorem: an extension of the notion of quasi-convexity

Abstract

Subject to suitable boundary conditions being imposed, sharp inequalities are obtained on integrals over a region Ω of certain special quadratic functions f ( E ), where E ( x ) derives from a potential U ( x ). With E =∇ U , it is known that such sharp inequalities can be obtained when f ( E ) is a quasi-convex function and when U satisfies affine boundary conditions (i.e. for some matrix D , U = D x on ∂ Ω ). Here, we allow for other boundary conditions and for fields E that involve derivatives of a variety orders of U . We define a notion of convexity that generalizes quasi-convexity. 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 quasi-convex quadratic functions. We also treat integrals over Ω 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 3×3 matrix-valued field satisfying ∇⋅ J =0.