Why is the laplacian of u zero for solutions to boundary problems?

[calvino]

Why does the laplacian of u=0 when u is a solution to a certain boundary problem? Is this always the case?

 

[calvino]

after working it out, I realized I was on a totally wrong track. It's simply dependent on the stipulations on the laplacian. Perhaps one can help me with one more thing. Are there any special facts about the laplacian of a solution to a pde problem?

 

[mathphys]

Lu=0 is the laplace equation (L is the laplacian operator), which is itself a PDE. A Function u which satisfies the laplace equation is called an harmonic function. If the solution of a different PDE satifies also the laplace equation (it's laplacian is zero) that solution is itself an harmonic function or a sum of them or may be expresed as a series of harmonic functions, because the laplacian is a linear operator and it obbeys the superposition principle. Of course this is a very informal explanation, but i think that is what you are asking for,, maybe

 

[Feynman]

Calvino, spesify your question?