Consider a boundary given by vertices (0,a), (0,0) and (1,0) (an 'L' shaped boundary).

The problem is to find the equation that passes between the endpoints (0,a) (1,0) of minimum length that encloses a specified area A.

A trivial case would be A=a/2 in which case the solution would be a line.

This is one dimensional Laplace problem with two boundaries (area and verticies) but how do I try to get a series solution for A