References for Green functions of $\nabla \cdot a \nabla$ on a domain with $a \in L^\infty$

Question

I am looking for a reference for basic properties of the Green function for a symmetric, uniformly elliptic operator $\nabla \cdot a \nabla$ where the coefficients $a_{ij}= a_{ji}$ are only assumed to be in $L^{\infty}$ and my domain is either $C^1$ or convex subset of $\mathbb R^n, n \ge 2$. Moreover, I am specifically thinking about the Dirichlet boundary condition.

I am looking for integrability/regularity estimates, symmetry, and some other basic statements.

I have done some analysis courses during my graduate studies, but it is not my first area. So when I was looking for such information all the books I found ditched the Green function approach when you leave the first few smooth examples of PDE's. I was wondering how similar (or different) such functions are when compared to the standard Laplacian case.

I appreciate any comments or references, thank you in advance.

Answer 1

This mathoverflow question has the references you are looking for. Namely, you can look in the original papers:

Littman, W.; Stampacchia, G.; Weinberger, H. F., Regular points for elliptic equations with discontinuous coefficients, Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 17, 43-77 (1963). ZBL0116.30302.

Grueter, Michael; Widman, Kjell-Ove, The Green function for uniformly elliptic equations, Manuscr. Math. 37, 303-342 (1982). ZBL0485.35031.Gruter-Widman '82.

A more recent reference, in a more general setting, is

Hofmann, Steve; Kim, Seick, The Green function estimates for strongly elliptic systems of second order, Manuscr. Math. 124, No. 2, 139-172 (2007). ZBL1130.35042.

It could be nice to look at the more modern presentation there and in the papers that cite it.

Life is wonderful when you have the Green's function in hand because you can use the representation formula to easily derive many regularity or decay estimates on solutions from the regularity or decay properties of the Green's function. On the other hand, it takes a bit of machinery to set up the Green's function, and in my experience, it is not always easy to locate what I need about it in the literature. Moreover, for certain systems (such as the Stokes equations in the half-space), the kernel can be unwieldy. For practical reasons, it can be quite useful to know how to prove the regularity or decay properties you want directly, without resorting to Green's functions. For example, one can compare the potential theoretic proof of Schauder's estimates (see Gilbarg-Trudinger), in which one perturbs off of the Green's function for the Laplacian by `freezing the coefficients', to Campanato's approach by energy methods. The latter generalizes easily to systems, while the former does not.