3
$\begingroup$

What is the explicit formula for a fundamental solution of the wave operator, where the space variable is in $\mathbb{R}^n$ with $n>1$? Thanks.

The operator I'm talking about is $$L=\partial_{tt}-\Delta_x$$ and for fundamental solution I mean a distribution $E$ (temperate) which satisfies $LE=\delta$ where $\delta$ is the Dirac distribution.

For $n=1$ one of such $E$ is $E(x,t)=(1/2)H(t)H(t^2-x^2)$ where $H$ is the Heaviside function.

I'm looking for expressions in higher dimensions.

  • 1
    You need to give more details. If you don't reveal an explicit description of "the" wave operators (except that operators do not have solutions, fundamental or otherwise -- equations do), you can't expect to get any explicit formulas in response.2012-06-28
  • 0
    Does $u=E$? Is delta some specific function or arbitrary or what?2012-06-28
  • 1
    Have a look here for starts: http://www.math.ucsd.edu/~lindblad/110b/l17.pdf http://www.math.ucsd.edu/~lindblad/110b/l18.pdf2012-06-28

3 Answers 3