Green's function of an operator

Question

Given a differential operator $L$, how is its Green's function defined?

I know that for a an initial condition problem it is the function so that the solution is defined by $u = G*f$, but I couldn't find a clear definition for an operaor.

Thanks.

Answer 1

Green's functions are often used in the context of boundary value problems, in which case their definition is specific to the boundary values: one has Dirichlet Green's function and Neumann Green's function, for example.

The one situation when we do not have boundary values is when the domain of consideration is all of $\mathbb R^n$. Wikipedia correctly notes that

If the kernel of $L$ is non-trivial, then the Green's function is not unique. However, in practice, some combination of symmetry, boundary conditions and/or other externally imposed criteria will give a unique Green's function.

(Also, in this case Green's function is known as the fundamental solution).

For example, consider the operator $Lf=f''$ on $\mathbb R$. By definition, Green's function with pole at $s$ is a function such that $LG =\delta(x-s)$. This means $f'$ must jump by $1$ at $s$ and be constant otherwise. One such function is $G(x,s)=\max(0,x-s)$, which I prefer to use as Green's function. But some prefer the symmetric function $G(x,s)=\frac12|x-s|$.

In higher dimensions, radial symmetry simplifies computations a lot, and nobody would consider anything other than $|x|^{2-n}$ (times some constant) as Green's function for the Laplacian in $\mathbb R^n$, $n>2$.

But on a noncompact manifold that is not as symmetric as the Euclidean space there may be no canonical choice for Green's function at all: no reason to prefer one over another.