Sharp inequalities that generalize the divergence theorem
Inverse Problems and Applications
08 April 14:00 - 15:00
Inverse problems often address the problem of what can be said about fields in the interior of a body from boundary measurements. The well known divergence theorem is often an essential tool: it gives an exact value for the integral over a region of the trace of E where E is the gradient of U. Subject to suitable boundary conditions being imposed we obtain sharp inequalities on integrals over a region of certain special quadratic functions f(E) where E derives from a potential U. When E is the gradient of 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. 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 a region 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 is a 3 by 3 matrix valued field with zero divergence. In that paper the result was used to obtain sharp bounds on the size of an inclusion in a three-dimensional body from Electrical Impedance Tomography measurements at the boundary.