# Definition of the weak solution to the Dirichlet boundary value problem

Let $\Omega \subset \mathbb{R}^n$ be open. The dirichlet problem is given by

\begin{align} -\Delta u &= f \text{ on } \Omega \\ u&=0 \text{ on } \partial \Omega \end{align}

for a function $f \in L^2(\Omega)$. When defining the weak solution I always saw that it was assumed that $\Omega$ is bounded and then the weak solution was defined as a function $u \in H^1_0(\Omega)$ such that $$\int \langle \nabla u, \nabla w \rangle = \int f w \text{ for all } w \in H^1_0(\Omega).$$

But does this definition still hold or even make sense if we do not assume that $\Omega$ is bounded, just that it is open?

• I think you mean $\int \nabla u \nabla w = \int fw$. – Lorenzo Quarisa Jul 3 '18 at 8:23
• oh yes thank you. But does it make sense to allow $\Omega$ to not be bounded? – vaoy Jul 3 '18 at 8:23

Yes, this definition can still be used for unbounded domains. For example, it appears in the paper Dirichlet problem for a linear elliptic equation in unbounded domains with $L^2$-boundary data by Chabrowski (this is just one of search results).