Estimates on the Green function of an elliptic second order differential operator. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T20:47:21Z http://mathoverflow.net/feeds/question/69521 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69521/estimates-on-the-green-function-of-an-elliptic-second-order-differential-operator Estimates on the Green function of an elliptic second order differential operator. semyon alesker 2011-07-05T08:00:28Z 2011-12-16T17:34:04Z <p>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:</p> <p>(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);</p> <p>(2) $G$ is infinitely smooth outside of the diagonal;</p> <p>(3) $G$ is bounded from below.</p> <p>The last property can be asked in a stronger form:</p> <p>(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.</p> <p>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.</p> http://mathoverflow.net/questions/69521/estimates-on-the-green-function-of-an-elliptic-second-order-differential-operator/83633#83633 Answer by semyon alesker for Estimates on the Green function of an elliptic second order differential operator. semyon alesker 2011-12-16T16:22:14Z 2011-12-16T16:22:14Z <p>I failed to find an explicit reference till now. However the result seems to be true. In the appendix to <a href="http://arxiv.org/PS_cache/arxiv/pdf/1111/1111.0403v2.pdf" rel="nofollow">this paper</a> we with a co-author have written it down (modulo some fact we could fid in the literature, mostly in Shimakura's book).</p> http://mathoverflow.net/questions/69521/estimates-on-the-green-function-of-an-elliptic-second-order-differential-operator/83641#83641 Answer by timur for Estimates on the Green function of an elliptic second order differential operator. timur 2011-12-16T17:34:04Z 2011-12-16T17:34:04Z <p>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.</p>