# Existence theorem for Laplace's equation.

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?

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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 $\partial\Omega$.

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Hi, Thanks for the response. Can you elaborate more on what is meant by smoothness. I would like a mathematical criteria vague terms like 'not so bad' and 'smooth' aren't of much help! – guru May 25 '13 at 17:14
@guru: 'Smooth' is not a vague term. It means the boundary can locally be given by the graph of an infinitely differentiable function. I suggest you to read an introductory PDE book. Better still, take a beginning graduate PDE course. The Dirichlet problem is among the first things covered in such a course. – timur May 27 '13 at 0:22

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'.

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Thanks. What about the Dirichlet problem? I don't really have access to that book. – guru May 25 '13 at 12:44
If you have $$\Delta u=0, u=g,$$ The first step is to translate this problem to $$\Delta u= F, u=0.$$ Now you can apply Lax-Milgram Lemma in a straightforward way. – guacho May 25 '13 at 12:49
Evans's book is a very good book, but in any first course book about PDE's should be a chapter concerning elliptic problems. Do you have access to the book of Gilbarg and Trudinger? – guacho May 25 '13 at 12:54
No. So basically are you saying that if I can find a function f, such that f = g on the boundary, then a solution of Laplace's equation will exist with those boundary conditions? – guru May 26 '13 at 8:22
Well, if f is in the correct space ($H^{-1}$) then you can translate your (original) problem to another problem with homogeneous Dirichlet boundary conditions and then you can apply Lax-Milgram. – guacho May 26 '13 at 17:22