Applications and motivations of resolvent for elliptic operator

Question

Let $ A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ be a matrix satisfying ellipticity condition, that is \begin{align} \mu^{-1}|\xi|^2\geq \sum_{i,j=1}^da_{ij}(x)\xi_i\xi_j\geq\mu|\xi|^2 \end{align}
with $ \mu>0 $ being a positive constant and $ \xi\in\mathbb{R}^d $. Since the spectrum of $ L=-\operatorname{div}(A(x)\nabla) $, for $ \zeta\in\mathbb{C}\backslash\mathbb{R}_+ $, $ (L-\zeta I)^{-1} $ exists. I read some paper and know that for some special $ A $ and $ p $, we can derive that \begin{align} \left\|(L-\zeta I)^{-1}\right\|_{L^p\to L^p}\leq \frac{C}{1+|\zeta|}. \end{align} where $ C $ may depends on $ \arg\zeta $. I want to know what is the motivation to study such estimates and where can I use such estimates.

Answer 1

To begin with, the ellipticity condition is useless if you don't ask also that $$\sum_{i,j}a_{ij}\xi_i\xi_j\le M|\xi|^2$$ for some finite constant $M$.

Now the resolvant estimate is used to define an operator $e^{-zL}$ for $\Re z>0$. In particular $S_t=e^{-tL}$ is the semi-groups associated with the evolution equation $$\partial_tu+Lu=f$$ where $f(t,x)\in L^1(0,T;L^p)$ is given, as well as an initial value $u_0\in L^p$.