= Constant-drift massive-Laplacian Dirichlet problem
Let $U\subset\mathbb R^d$ be bounded, $A\in\mathbb R^d$ satisfy $\lVert A\rVert_2<2$, and $f\in L^2(U)$. The <bilinear form>
$$
B(u,v)=\int_U\bigl(\nabla u\mathbin\cdot\nabla v+uv+uA\mathbin\cdot\nabla v\bigr)
$$
on $H_0^1(U)$ is bounded. The <Cauchy-Schwarz inequality> and $2ab\leq a^2+b^2$ give
$$
B(u,u)\geq\left(1-\frac{\lVert A\rVert_2}{2}\right)\lVert u\rVert_{H^1}^2,
$$
so it is a <coercive bilinear form>. The <Lax-Milgram theorem> therefore gives a unique <weak solution> of $-\Delta v+v+A\cdot\nabla v=f$ with zero <Dirichlet boundary condition>.
Back to article page
