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It is well-known that $u(x,y,t)=(4\pi t)^{-n/2}(e^{-|x-y|/4t}+e^{-|x-y'|/4t})$, $x,y\in \mathbb{R}^n_+=\{x\in \mathbb{R}^n|x_n\geq 0\}$, $y'=(y_1,\dots,y_{n-1},-y_n)$, is Neumann heat kernel of $\partial_tu-\Delta_yu=0$ in $\mathbb{R}^n_+$, $\partial_{y_n}u=0$ on $\partial\mathbb{R}^n_+$.

Is there any other explicit example of Neumann heat kernels?

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    $\begingroup$ A quarter-plane and analogues in higher dimensions.Also in one dimension on a segment as series. $\endgroup$
    – Andrew
    Commented Apr 12, 2019 at 15:59
  • $\begingroup$ Thanks Andrew. However, I would like to see explicit examples in the higher dimension, $B(x,R)\subset\mathbb{R}^n$ or hemisphere, for example. I don't know if that's possible. $\endgroup$
    – Truong
    Commented Apr 13, 2019 at 0:55

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