Principal fundamental solution

A fundamental solution $G(x,y)$, defined throughout the space $E^n$, of a second-order elliptic equation

$$Au=\sum_{i,k=1}^na_{ik}\frac{\partial^2u}{\partial x_i\partial x_k}+\sum_{i=1}^nb_i(x)\frac{\partial u}{\partial x_i}+c(x)u=0\tag{*}$$

that satisfies the conditions

$$G(x,y)=o(e^{-a|x-y|}),\quad\frac{\partial G}{\partial x_i}=o(e^{-a|x-y|})$$

for certain positive constants $a$ and $R$ if $|x-y|>R$.

If the coefficients $a_{ik}(x)$, $b_i(x)$ and $c(x)$ satisfy a Hölder condition on $E^n$ and if the inequality $c(x)<-\gamma$ is satisfied for some $\gamma>0$, then a principal fundamental solution exists. If the coefficients of the operator $A$ are defined in a certain bounded domain with smooth boundary, then they can be extended to the entire space $E^n$ so that a principal fundamental solution will exist for the extended operator.

References

[1]	C. Miranda, "Partial differential equations of elliptic type" , Springer (1970) (Translated from Italian)