Suppose we have a second order elliptic differential operator
$$
L(v) = -\text{div}(A(x) \nabla v)
$$
$A(x)$ is a bounded and strictly positive definite matrix with Hölder continuous entries. And suppose $\Omega$ is a $C^1$ domain (can be considered more regular if required).

From [2], Theorem 1, we know that there exists a Poisson Kernel $K(x,y)$ for the above operator and domain $\Omega$. That is the solution $u$ such that $Lu=0$ in $\Omega$ can be written as
$$
u(x) = \int_{\partial \Omega}u(y) K(x,y) d\mathcal H^{N-1}(y)
$$

On the other hand, we have from [1] we have the existence of Green's function $G$ for the operator $L$ in the domain $\Omega$. Natural question to ask is just like we have a integral representation for harmonic functions, can we say that the Poisson Kernel $K(x,y)$ and the co-normal derivative of Green's function on the boundary i.e. $\left ( A(y)\nabla_y G(x,y)\right )\cdot \nu $ are equal on $\partial \Omega$?

In particular, since the Green's function $G$ of $L$ exists for the domain $\Omega$, then for $u$ such that $Lu=0$ is the following expression true?
$$
u(x) = \int_{\partial \Omega}u(y) \left ( A(y)\nabla_y G(x,y)\right )\cdot \nu_y\,d\mathcal H^{N-1}(y).
$$

The above claim hods true when $A(x)= Id$, i.e. when the operator $L$ is simple Laplacian. Under what condition/s we can say the same for a general second order elliptic operator in divergence form?

Article
The Green function for uniformly elliptic equations

If you plan to formulate more than one problem all sharing the same background (e.g. they are all from the same paper) then please choose "Group", otherwise select "Single" option.

No remarks yet