On the uniqueness of the solution of static Maxwell equations. (Q1432206)

The electrostatic field in a medium obeys the equations \[ \text{rot\,}\vec E= 0,\quad \text{div\,}\vec D= 4\pi\rho\tag{1} \] in which \(\rho\) is the density of external charges and the relation between \(\vec E\) and \(\vec D\) is specified by the constitutive equation \[ F(\vec E,\vec D)= 0.\tag{2} \] Vector pairs \(\{\vec E,\vec D\}\) satisfying the constitutive equation (2) are called CE points (they form a point set in the six-dimensional \(\{\vec E,\vec D\}\)-space); two different CE points correspond to the inequality \[ (\Delta\vec E)^2+ (\Delta\vec D)^2\neq 0.\tag{3} \] Accordingly, any solution \(\{\vec E(\vec r),\vec D(\vec r)\}\) to the problem can be viewed as a point of the \(\vec r\)-space associated with a CE point. The following uniqueness criterion is presented: If any two distinct CE points satisfy the strict inequality \[ \Delta\vec E\Delta\vec D> 0\tag{4} \] then the solution to the electrostatic problem in such a medium is unique.