Stability (wrt parameters) of elliptic partial differential equation consider the equation 
$$\mathcal Lu=f \quad \text{in  } \Omega $$
With some appropriate boundary condition, $\Omega$ regoular as you like,   $ \mathcal  L$ to be defined by
$$\mathcal Lu=-div(A(x)\nabla u ) +div(\vec bu)+\vec c\nabla u + ru $$
Is there any stability extimate of the solution when the parameters vary?
For example, using a weak formulation, one could prove that 
$$\text{if} \quad  A^k \rightarrow A  \quad (L^{ \infty} \text{norm should work) then:} \quad u^k \rightarrow u \quad \text{$u^k$ being the solution with $A^k$} $$
 A: Let's assume three things.  First, that the "appropriate boundary condition" is $u=0$ on $\partial \Omega$.  Second that we write $L_k$ for the operator defined with $A_k,b_k,c_k, r_k$.  Third, we need to avoid the possibility that $0$ is in the spectrum of the operator $L_k$ for any $k$ so that we can get estimates for solutions to $L_k u =f$ solely in terms of $f$.  Let's assume this.
Now suppose that $L_k u_k = f \in H^{-1}$ for each $k \ge 1$.  Then the usual $H^1(\Omega)$ a priori estimate for weak solutions shows that 
$$
\| u_k \|_{H^1_0} \le C(\|A_k,b_k,c_k,r_k\|_{L^\infty},\Omega) \|f\|_{H^{-1}}
$$
where $C$ can be shown to increase with respect to the $L^\infty$ part.  Consequently, if $(A_k,b_k,c_k,r_k) \to (A,b,c,r)$ in $L^\infty$ and $A$ remains uniformly elliptic on $\Omega$, then 
$$
\sup_{k\ge 1}\| u_k \|_{H^1_0} \le \tilde{C}(\|A,b,c,r\|_{L^\infty},\Omega) \|f\|_{H^{-1}}
$$
for some other constant $\tilde{C}$.  
We can now use this bound together with weak compactness and Rellich's  theorem to extract a subsequence (still denoted $u_k$ for the sake of simplicity) such that $u_k \rightharpoonup u$ weakly in $H^1_0$ and $u_k \to u$ strongly in $L^2$.  Then 
$$
\int_\Omega A_k \nabla u_k \cdot \nabla v - u_k b_k \cdot  \nabla v + c_k \cdot \nabla u_k v + r_k u_k v   = <f,v>
$$
for all $v \in H^1_0(\Omega)$, and so we can send $k \to \infty$ and use the above convergence results to deduce that 
$$
\int_\Omega A \nabla u \cdot \nabla v - u b \cdot  \nabla v + c \cdot \nabla u v + r u v   = <f,v>
$$
for all $v \in H^1_0$, and hence $u$ solves 
$$
-\text{div}(A \nabla u) + \text{div}(bu) + c\cdot \nabla u + ru =f.
$$
Note: without the condition of avoiding $0$ we would have to include the $L^2$ norm of $u_k$ on the right side of our basic a priori estimate, and so we would have to have boundedness in $L^2$ in order to extract limits.
