(1)    Absolute value of tension is constant on the boundary of an optimal hole.
Remark. Notice that here we are dealing with a free boundary problem: The boundary of the domain is not specified; instead, the additional requirement (1) is posed.


Next, we find the geometrical shapes that satisfy the condition  (1) .

A. Loadings of the same sign

A hole that has a constant tension on its boundary satisfies equation (1)
It is not too difficult to guess solutions of this type:

The circle is optimal if the loading is even from all directions ( V=H) . In this case, a=1.


The ellipse is optimal if the loading is uneven ( 0<H< V) .In this case, 0 < a < 1.

The optimal ellipse is oriented so that its smaller axis is parallel to the direction of minimal loading
and the ratio of its axes is equal to a.

A crack-like inclusion is optimal if the loading is applied from only one side ( 0=V< H). In this case, a = 0.

So far, the problem has an elegant classical solution...

To the third part