Solvability of Laplace’s Equation

We’ve shown that if a solution exists, it’s unique (though only up to an additive constant for the Neumann boundary condition; in this case there’s also a requirement on g). We’ve also shown other interesting properties possessed by any solution, e.g., the maximum principle and the mean value property. Existence is actually the toughest issue. On certain special domains like rectangles and circles/spheres it becomes much easier. We’ll look at those first, then at the more general problem.