Same perimeter and area for a circle and an ellipse For a given circle, is there exist an ellipse with same perimeter and area as to that circle? 

If not, that is my suspicion, is in three-dimension parallel question: For a given sphere, is there exist an ellipsoid with same surface area and volume as to that sphere?
No, because the isoperimetric quotient $A/P^2$ is smaller for a non-circular ellipse than for any circle.
I cannot give you a deep rigourous argument, but surface tension is the reason a soap-bell is spherical so...

A circle has a specific eccentricity, and you choose another eccentricity while the area is fixed, there are no more parameters left to adjust for the perimeter.