"Jump" boundary condition, or a Periodic Boundary Condition with offset

Hi all,

I am interested in solving the Laplace equation with a particular set of boundary conditions.

Normal periodic boundaries impose
u at Border A = u at Border B.

I want to impose
u @ Border A = u @ Border B + 1

I would have to impose a Lagrange multiplier to specify what the function integrates to over all space, but would this be doable?

In this way, it creates a gradient throughout the system while not constraining the solution at the edge to be a constant or a set function like in the Dirichlet boundary conditions.

Has anyone done this before? If so, do you have any tips?