the Neumann compatibility condition

If you seal a room so that heat can only enter or leave through one wall, and you keep pumping heat in faster than it leaves, the room will never settle to a steady temperature — it just keeps warming. Steady state is possible only if the heat coming in exactly balances the heat going out. The Neumann compatibility condition is the precise statement of this balance, and it is the requirement that must hold before the Neumann problem can have a solution at all.

Precisely, for Poisson's equation Laplacian u = f on a region D with prescribed boundary flux du/dn = g, integrate the equation over D and apply the divergence theorem. The integral of Laplacian u over D equals the integral of du/dn over the boundary, so the data must satisfy: the integral of g over the boundary equals the integral of f over D. For Laplace's equation (f = 0) this collapses to the integral of g over the boundary being zero — the net outward flux must vanish. If the data violate this, no solution exists; if they satisfy it, a solution exists and is unique up to an additive constant.

This is a textbook example of a solvability condition for a problem whose operator has a non-trivial null space — here the constants, which the Laplacian annihilates and whose normal derivative is zero. The Fredholm alternative organizes exactly this: either the problem is uniquely solvable, or it is solvable precisely when the data are orthogonal to the null space, with the solution then non-unique by that null space. The Neumann problem is the second case. Honest practice means always checking compatibility before you try to solve — a numerical Neumann solver fed incompatible data will either fail to converge or silently return garbage.