# Can be the duality theorem applied to not locally non-satiated utility functions?

I have the following not locally non-satiated utility function:

$$U(x,y)=-(x-1)^2-(y-2)^2$$

where $U(x,y): \, \!R^n_+ \rightarrow \!R$

The 3D plot of this function is an infinite paraboloid; therefore, $U(x,y)$ can be considered as a not locally non-satiated utility function.

After I've found the Marshallian demand and the indirect utility functions, I'd like to determine the Hicksian demand curve and the expenditure functions using the duality theorem. Can be the duality theorem applied to this type of not locally non-satiated utility functions?