I am looking for a potential theory representation for the following equation in $2$D:

$$\vec{\nabla} \cdot \left(a(x) \vec{\nabla}u\right) = 0 \,\, \forall x \in \Omega \,\, (\spadesuit)$$ $$u = g \,\, \forall x \in \partial \Omega$$

If $a(x)$ were to be a constant, then from here, I find that the solution can be represented as $$u(x) = \int_{\partial \Omega} \left(\vec{\nabla}_x G(x-y) \cdot \hat{n}\right) \sigma(y)dy \,\,\, \forall x \in \Omega$$ where $G(x-y)$ is the free-space Green's function, $\log(x-y)$ and $\sigma(y)$ satisfies the linear system $$-\dfrac{\sigma(x)}2 + \int_{\partial \Omega} \left(\vec{\nabla}_xG(x-y) \cdot \hat{n}\right) \sigma(y)dy = g(x) \,\,\, \forall x \in \partial \Omega$$ From this I can solve for $\sigma$ on the boundary and reconstruct my solution $u(x)$ in the domain.

Since for my problem $a(x)$ is not a constant, I introduced a volume potential and represented my solution as $$u(x) = \int_{\Omega} G(x-y) \psi(y) dy + \int_{\partial \Omega} \left(\vec{\nabla}_xG(x-y) \cdot \hat{n} \right) \sigma(y)dy \,\,\, \forall x \in \Omega$$

The unknowns now are $\psi(y)$ in the domain and $\sigma(y)$ on the boundary.

Applying the boundary condition, I obtain $$\int_{\Omega}G(x-y)\psi(y) dy-\dfrac{\sigma(x)}2 + \int_{\partial \Omega} \left(\vec{\nabla}_xG(x-y) \cdot \hat{n}\right) \sigma(y)dy = g(x) \,\,\, \forall x \in \partial \Omega$$

When I plug in my representation into the PDE $(\spadesuit)$, I have

$$a(x) \psi(x) + \vec{\nabla} a \cdot \int_{\Omega} \vec{\nabla} G(x-y)\psi(y) dy + {\color{blue}{a(x) \nabla^2 \left(\int_{\partial \Omega} \left(\vec{\nabla}_xG(x-y) \cdot \hat{n}\right) \sigma(y)dy\right)}} + $$ $$\color{red}{\vec{\nabla} a(x) \cdot \left(\int_{\partial \Omega} \vec{\nabla}\left(\vec{\nabla}_xG(x-y) \cdot \hat{n}\right) \sigma(y)dy\right)} = 0 \,\,\, \forall x \in \Omega$$

My questions beyond this point are the following:

  1. ${\color{blue}{\text{The blue term}}}$ vanishes to zero, since $x \notin \partial \Omega$.

  2. Can ${\color{red}{\text{the red term}}}$ be simplified further?

  3. Also, can someone suggest a reference, which discusses different representations of solutions for different elliptic PDE's with examples?


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