the Neumann problem
/ NOY-mahn /
Instead of fixing the temperature on the edge of a plate, suppose you control how much heat flows in or out across the edge — you insulate parts of it, or pump heat through at a chosen rate. You then ask what equilibrium temperature this produces inside. That is the Neumann problem: prescribe the boundary flux rather than the boundary value.
Precisely, the Neumann problem for Laplace's equation seeks u with Laplacian u = 0 inside the region and a prescribed value of the outward normal derivative on the boundary, written du/dn = g, where du/dn is the rate of change of u as you step out across the boundary. (For Poisson's equation the interior condition is Laplacian u = f.) Physically du/dn is the heat flux, the electric field component, or the normal flow velocity at the boundary. Two features distinguish it from the Dirichlet problem. First, the solution is unique only up to an additive constant: since only the gradient of u is constrained, you can raise the whole solution by any constant and it still satisfies everything — the absolute level is undetermined. Second, you are not free to prescribe g arbitrarily.
That second point is the compatibility condition: by the divergence theorem the total flux through the boundary must equal the total source inside, so the boundary data g must integrate to the right amount (for Laplace's equation, to zero). Physically, if the region is insulated except where you pump heat, the net heat you pump in must be zero in steady state — otherwise the temperature would keep rising and no equilibrium exists. The Neumann problem models insulated or flux-driven equilibria: a charged conductor with prescribed surface field, an insulated heat sink, the flow past a body where the normal velocity is set by the body's shape.
Insulate every edge of a plate (du/dn = 0 all around) with no internal source. The compatibility condition is satisfied (total flux 0), and the only solutions are the constants: with no heat entering or leaving, every uniform temperature is a steady state, which is exactly the up-to-a-constant non-uniqueness.
Flux data fix the shape but not the level; they must also balance, or no steady state exists.
Two genuine differences from the Dirichlet problem trip people up: Neumann solutions are unique only up to a constant, and the boundary data cannot be chosen freely — they must satisfy the compatibility (solvability) condition. Forget either and you will look for a solution that does not exist or expect a uniqueness you will not get.