Confirming the Existence the PDE Solutions in Sobolev Spaces

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 ....

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 $u=v$.

• Lax-Milgram theorem also gives uniqueness. It is not needed to prove it separately. Dec 14, 2016 at 3:12
• You are right. In fact, I simply like this proof of uniqueness since it is easy to write and uses several useful estimates. However, I will edit my answer to indicate that Lax-Milgram is sufficient. Thank you!
– Matt
Dec 14, 2016 at 3:13
• Yes, it is instructive to prove it separately (especially because in general the uniqueness is not easily given with the existence, which happens if we use the Galerkin method as sketched here). Dec 14, 2016 at 3:31
• So, if $f \in H^{-2}$, then to maintain integrability, I need to go up to $u \in H^2$, right? Now what happens it I need am using a different $p$-norm? Dec 15, 2016 at 4:46
• If $f\in H^{-2}$ you can expect the solution to be $u\in L^2$ but your test functions $v$ would come from $v\in H^{2}_0$. I do not know anything interesting to say about different $p$-norms, but I can suggest for example looking up literature on weak solutions of the $p$-Laplacian. There you can expect solutions in $W^{1,p}$.
– Matt
Dec 15, 2016 at 14:26