Periodic Boundary Conditions for a 2D Problem

Hi, I am new to FreeFem++, and I am trying to solve a BVP to obtain the micropolar strain localization tensor in a 2D square with some periodic boundary conditions that its dimensions are as follows: [-0.5,0.5]×[-0.5,0.5]. I have four unknowns, h1,h2,h3, and h4.

The boundary conditions are
h1(-0.5,y)+ h3(-0.5,y)= h1(0.5,y)+ h3(0.5,y)
h2(-0.5,y)+ h4(-0.5,y)= h2(0.5,y)+ h4(0.5,y)
h1(x,-0.5)+ h3(x,-0.5)= h1(x,0.5)+ h3(x,0.5)
h2(x,-0.5)+ h4(x,-0.5)= h2(x,0.5)+ h4(x,0.5)

I have a problem with defining these boundary conditions in FreeFem++.

I would appreciate it if you could help me with this.

If you problem is linear make a change of variable

h13 = h1+h3
h31 = h1-h3
h24 = h2+h4
h42 = h2-h4
h13 and h24 will be real periodic,
which BC on h31 and h42 ?

if you know ? you can write the Variationnal form.
otherwise you problem is not well defined!

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Dear Prof. Hecht,

Thank you for your help, it worked well for my problem.