Let $L$ be an elliptic operator on $\Bbb C$ with $$\DeclareMathOperator{\Img}{Im} L^{-1}f(z)=\int_{\Bbb {C}}K_1(|z-w|^2)f(w) e^{i\cdot\Img\langle z,\overline{w}\rangle} \, dw $$ where $K_1$ is the modified Bessel function. I want to define the kernel Poisson to the Dirichlet problem $Lu=0 $ on the unit Ball $B$ of $\Bbb C$ with $u=f$ on the sphere $S$. How can I do that? Thank you in advance.
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$\begingroup$ Related, but dealing with the construction of fundamental solution instead of the construction of the Green function. Remember that once you have a fundamental solution, you can construct a Green function for a given domain if you know an entire solution for the same operator $\endgroup$– Daniele TampieriCommented Aug 17 at 11:51
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$\begingroup$ @DanieleTampieri Can you explain in more detail the last part? $\endgroup$– Giorgio MetafuneCommented Aug 17 at 13:30
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1$\begingroup$ The notation and terminology here is a bit confusing. I think that the starting point to match the OP's question to Daniele's discussion is the fundamental solution $E(z-w) \text{ "=" } K_1(|z-w|^2) e^{i\operatorname{Im} z\cdot\bar{w}}$, which already doesn't work because the rhs is not translation invariant, so one would have to write the lhs as $E(z,w)$. Next, the correct interior Dirichlet Green function is $\mathscr{G}(z,w) = E(z,w) - G(z,w)$, where the crucially necessary $G(z,w)$ is the unknown "entire solution". The "Poisson kernel" is then the differentiated $\mathscr{G}$. $\endgroup$– Igor KhavkineCommented Aug 18 at 10:41
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1$\begingroup$ @IgorKhavkine. Thank you for your help. $\endgroup$– Ryo KenCommented Aug 18 at 23:00
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1$\begingroup$ My last remark is that the reformulation of the problem in terms of the "entire solution" or the interior Dirichlet Green function leaves the OP none the wiser, as both reformulations are equally hard and are at least as hard if not harder than finding the desired "Poisson kernel". $\endgroup$– Igor KhavkineCommented Aug 20 at 19:56
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