Confirming the Existence the PDE Solutions in Sobolev Spaces

Question

So far I have only the most basic understanding of the Sobolev space, like the existence of a unique weak solution, or what it means in the PDE: $$\Delta u = -f ,\qquad \text{where } f \text{ is a Schwartz function}$$ However, how do I think about a PDE when $f$ only satisfies a weaker condition, like $f \in L^2$. I think I got lost in all the different kinds of Sobolev spaces, and what is allowed in each space. So, just take that $f \in L^2$. Can you kindly show how you examine the existence of a unique solution to the Laplace equation above? Thanks ....

Answer 1

I assume that $\Omega$ is bounded and has smooth boundary and that you have zero Dirichlet boundary conditions. If $f\in L^2(\Omega)$ (in fact one can take weaker conditions $f\in H^{-1}(\Omega)$ here) then consider the weak formulation of the Poisson equation:

$$\int \nabla u \cdot \nabla v = \int f v, \forall v\in H^1_0(\Omega).$$

Using the Lax-Milgram theorem, one can deduce existence and uniqueness of solutions $u \in H^1_0(\Omega)$ satisfying the weak formulation.

In my opinion, it is also instructive to establish uniqueness by hand since it requires several ubiquitous estimates. Note that

$$\|\nabla u \|^2_{L^2(\Omega)}=\int \nabla u \cdot \nabla u = \int f u \leq \|f\|_{L^2(\Omega)}\|u\|_{L^2(\Omega)}\leq C_\epsilon\|f\|_{L^2(\Omega)}^2+\epsilon \|u\|_{L^2(\Omega)}^2.$$

Using the Poincare inequality for $u$: $\|u\|_{L^2(\Omega)}\leq C\|\nabla u\|_{L^2(\Omega)}$ and taking $\epsilon$ sufficiently small we have

$$\|\nabla u\|_{L^2(\Omega)} \leq C \|f\|_{L^2(\Omega)}.$$

So, assuming there are two weak solutions $u,v\in H^1_0(\Omega)$, consider $w=u-v$ which solves $\Delta w = 0$ with $w=0$ on $\partial \Omega$. Our above estimate establishes that $w\equiv 0$ and so $u=v$.