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- #1

- Apr 13, 2013

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Let $a,b>0$ and $D$ the rectangle $(0,a) \times (0,b)$. We consider the boundary value problem in $D$ for the Laplace equation, with Dirichlet boundary conditions,

$\left\{\begin{matrix}

u_{xx}+u_{yy}=0 & \text{ in } D,\\

u=h & \text{ in } \partial{D},

\end{matrix}\right.$

where $h:[0,a] \times [0,b] \to \mathbb{R}$ given function.

Supposing that $h$ is equal to $0$ at the vertices of the rectangle, I want to prove that the solution of the problem is the sum of the solutions of four respective problems, with homogeneous Dirichlet boundary conditions in three sides of the rectangle.

If we suppose that $h$ is equal to $0$ at the vertices of the rectangle, don't we have a problem with homogeneous Dirichlet boundary conditions? Or am I wrong? So do we maybe have to suppose that $h$ is nonzero?