Elliptic PDE, uniqueness of solution

Question

I'm considering a partial differential equation of the form

$$\nabla^2 u + \mathbf{a}\cdot\nabla u = 0$$

with Dirichlet boundary conditions, where $\mathbf{a}$ is a (smooth, nonconstant) vector field. Can anybody give me a hint how to find sufficient conditions on $\mathbf{a}$ so that the solution is unique? Thank you.

Answer 1

For this equation, the weak maximum principle is satisfied. You just need $a\in L^\infty(\Omega)$. See Gilbarg/Trudinger, Chapter 8. The result you are looking for is Theorem 8.3

Answer 2

Let $w$ be difference of two solutions. Then $w$ satisfies the same equation with zero Dirichlet's boundary condition. Now multiplying the equation by $w$ integrating over the domain $D$ and using divergence theorem along with a few algebra you arrive at $$\int_D|\nabla w|^2dx+\frac{1}{2}\int_D(\nabla\cdot a )w^2dx=0.$$ In order to show that $w=0$, besides $a\in L^\infty(D)$ you simply need $\nabla\cdot a\geq 0$. Also you may apply the Poincare inequality to the first integral to find a term $\lambda_1^{-2}+\frac{1}{2}\nabla\cdot a$ (depending on the first eigenvalue $\lambda_1>0$) which will be the coefficient in front of $w^2$ in the integral should have definite sign in $D$.