Existence theorem for Laplace's equation.

Question

Regarding Laplace's equations, the following two are my questions: 1) If Dirichlet boundary conditions are specified on the surfaces, is it guaranteed that a solution will exist? Or is there some condition that the boundary conditions themselves must satisfy? 2) If Neumann boundary conditions are specified on the surfaces, is it guaranteed that a solution will exist? Or is there some condition that the boundary conditions themselves must satisfy?

Answer 1

You have $$ \Delta u= 0, \nabla u\cdot n=g, $$ then you need the following compatibility condition $$ \int_{\partial \Omega} g(s)ds=0. $$ For the Dirichlet problem I refer you the book of Evans 'Partial Differential Equations'.

Answer 2

1) For the Dirichlet problem you don't need compatibility condition, but you need that the boundary is not so bad. For instance, smooth surfaces will work.

2) For the Neumann problem, apart from some smoothness of the boundary, you need a compatibility condition that follows immediately from the divergence theorem (also known as integration by parts): $$ \int_\Omega \Delta u = \int_{\partial\Omega}\partial_\nu u, $$ where $\partial_\nu u$ is the outward normal derivative of $u$ at the boundary $\partial\Omega$.