Estimates on the Green function of an elliptic second order differential operator. - MathOverflow

Estimates on the Green function of an elliptic second order differential operator.

2011-07-05T08:00:28Z

Let $D$ be a linear differential elliptic operator of second order with infinitely smooth coefficients acting on real valued functions on a compact manifold $M$. Let us assume that $D$ has no free term, i.e. $D(1)=0$. Let us fix a smooth positive measure (density) $\mu$ on $M$. Does there exist a (integrable) Green function $G\colon M\times M\to \mathbb{R} $ with the following properties:

(1) $\int_M G(x,y) \cdot D\phi(y) d\mu(y) =\int_M\phi(y) d\mu(y) -\phi(x)$ for any function $\phi$ and $x\in M$ (this is the definition of Green function);

(2) $G$ is infinitely smooth outside of the diagonal;

(3) $G$ is bounded from below.

The last property can be asked in a stronger form:

(3') Does $G$ satisfy the asymptotic estimate near the diagonal: $$c|x-y|^{2-n}\leq G(x,y)\leq C|x-y|^{2-n}$$ where $c,C>0$ and $n=\dim M>2$. If $n=2$ there should be a logarithmic estimate.

I am pretty sure that this is true and should be well known. I would need a reference. The special case when $D$ is the Laplacian for a Riemannian metric on $M$ is contained explicitly in some textbooks I am familiar with.

Answer by semyon alesker for Estimates on the Green function of an elliptic second order differential operator.

2011-12-16T16:22:14Z

I failed to find an explicit reference till now. However the result seems to be true. In the appendix to this paper we with a co-author have written it down (modulo some fact we could fid in the literature, mostly in Shimakura's book).

Answer by timur for Estimates on the Green function of an elliptic second order differential operator.

2011-12-16T17:34:04Z

Green's functions are constructed in Aubin's book for operators such as you mentioned, but with some sign condition on the lowest order term. I have not had a close look but my suspicion is that (3') is fine but for (3) you need a maximum principle.