Reference request: Uniformly elliptic partial differential operator generates positivity preserving semigroup

Question

I am looking for a reference of the following result:

Let $\Omega\subset \mathbb{R}^n$ be be a bounded domain with smooth boundary. Let $$A = \sum_{i,j=1}^n \partial_i ( a_{ij} \partial_j) + \sum_{i=1}^n b_i \partial_i + c$$ with domain $$D(A) = H_0^1(\Omega) \cap H^2(\Omega)$$ and let $a_{ij}= a_{ji}, b_i\in W^{1,\infty}(\Omega)$, $i,j=1,\dots,n$, $c \in L^{\infty}(\Omega)$. Furthermore, assume that there is a $\eta>0$ such that $$\sum_{i,j=1}^n a_{ij} \xi_i \xi_j \geq \eta | \xi |^2$$ for all $\xi\in \mathbb{R}^n$. Then, $A$ generates a positivity preserving semigroup $T(t)$, $t\geq 0$, on $L^2(\Omega)$, i.e., $$T(t) f \geq 0$$ whenever $f\in L^2(\Omega)$, $f\geq 0$ almost everywhere.

I suspect that this is true, but I haven't been able to find a reference for this result. Does anyone know where to find this?

Thank you!

Answer 1

Check Chapter 4, more specifically Chapter 4.2 in Ouhabaz: Analysis of heat equations on domains (2005). ZBL1082.35003. The approach there goes by the form method which I personally perceive to be very transparent and overall elegant.

You will see that with real coefficients in the elliptic operator---which is implicitly assumed if you ask for positivity, I guess---it only remains to check some invariance properties for the form domain which happens to be $H^1_0(\Omega)$ in your case. (And then these are well known I think!) Let me also mention that you do not need a smooth domain or Lipschitz coefficients for positivity or many other properties if you are willing to dispose of the optimal domain $H^2(\Omega) \cap H^1_0(\Omega)$ for the $L^2$-operator.