In principle, I think the answer is yes, by introducing the simple disk as you have, which was a good idea. So for , you would let . For , the domain is a bit trickier, but you can replace with and make to get or . I think this might be what you are looking for.
However, keep in mind that in one dimension, there is little point to Lagrange multipliers. Consider: if , then by letting , , so , i.e. , which is the standard one-dimensional way of finding the maxima/minima anyway, having nothing to do with .
You might try experimenting with your method on functions over two variables to see if there is something truly new to be discovered.