I have the PDE problem

$$-\nabla^2u=0,\quad x\in\Omega$$ $$u=g,\quad x\in\partial\Omega$$

I have to figure out the integral equation involving the double layer potential for solving the above PDE. Now the boundary here is star shaped that it says can be written in polar coordinates as $r=R(\theta)$, $\theta\in[0,2\pi)$.

Since $u$ is harmonic, I started with writing the solution as a combination of the single layer potential of $\partial_\nu u$ and double layer potential of $u$ as follows

$$u(x)=\int_{\partial\Omega}\Phi(x-\sigma)\partial_\nu u(\sigma)d\sigma-\int_{\partial\Omega}g(\sigma)\partial_\nu\Phi(x-\sigma)d\sigma$$ where $\Phi$ is the fundamental solution. We get rid of the single layer potential here since we do not know $\partial_\nu u$. For that we forget about the single layer potential and try to represent $u$ in the form of a double layer potential $\mathcal{D}$ by choosing appropriate density. So we seek a continuous function $\mu$ on $\partial\Omega$, so that the solution $u$ is


This is harmonic in $\Omega$, so we check the boundary condition the jump relation

$$\lim_{x\to z\in\partial\Omega}u(x)=g(z)$$

Letting $x\to z\in\partial\Omega$, with $x\in\Omega$, and taking into account the jump relation

$$\lim_{x\to z,x\in\Omega}\mathcal{D}(z;\mu)=\mathcal{D}(z;\mu)-\frac{1}{2}\mu(z)$$

and thus we get the following integral equation for $\mu$ as

$$\int_{\partial\Omega}\mu(\sigma)\partial_\nu\Phi(z-\sigma)d\sigma-\frac{1}{2}\mu(z)=g(z),\quad z\in\partial\Omega$$

Now all this analysis is for a smooth domain. But here we have got a domain with star shaped boundary. How to change things accordingly? I do not know much about the analysis related to Lipschitz domain.


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