= Poisson convergence on expanding balls in a homogeneous Sobolev space
{c}
{title2=$v_R\to v\text{ in }\dot H^1(\mathbb R^3)$}
In three dimensions let $f\in L^{6/5}(\mathbb R^3)$ have <compact support>. Let $v_R\in H_0^1(B_R)$ solve $-\Delta v_R=f$ and extend it by zero. There is a unique $v\in\dot H^1(\mathbb R^3)$ with $\int\nabla v\cdot\nabla\phi=\int f\phi$ for all $\phi\in\dot H^1$. Indeed <Holder inequality> and the homogeneous <Sobolev inequality> make the right side a continuous <linear functional> in the <gradient> <norm>, and the <Riesz representation theorem> applies to this <Hilbert space>.
The <weak formulation> says that $v_R$ is the <orthogonal projection> of $v$ onto the closed subspace consisting of zero extensions of $H_0^1(B_R)$. These subspaces increase with $R$ and their union contains every <test function>, which is dense by the definition of the <homogeneous Sobolev space>. The best-approximation property of <orthogonal projection> therefore gives $\|\nabla(v_R-v)\|_2\to0$. The homogeneous <Sobolev inequality> also gives $\|v_R-v\|_6\to0$. This convergence does not generally extend to the inhomogeneous <Sobolev space> $H^1$, as the <spherical mean of a Poisson solution outside its source> shows for a source of nonzero total mass.
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