Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
add comment

2 Answers

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 $\partial\Omega$.

share|improve this answer
    
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
add comment

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'.

share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.