From Salsa, PDEs in Action, chapter 5.

Let $w_\phi$ be the solution of the Cauchy problem \begin{array}{l} w_{tt}-c^2 \Delta w=0 \quad x\in\mathbb{R}^3,\; t>0 \\ w(x,0) = 0 \quad x\in\mathbb{R}^3 \\ w_t(x,0)=\phi(x) \quad x\in\mathbb{R}^3 \end{array} such that $w\in C^3(\mathbb{R}^3\times[0,+\infty))$.

Then $v\equiv \partial_t w_\phi$ is the solution of the Cauchy problem \begin{array}{l} v_{tt}-c^2 \Delta v=0 \quad x\in\mathbb{R}^3,\; t>0 \\ v(x,0) = \phi(x) \quad x\in\mathbb{R}^3 \\ v_t(x,0)=0 \quad x\in\mathbb{R}^3 \end{array}

This lemma, combined with the superposition principle, simplifies the solution of the Cauchy problem with both Cauchy data.

I cannot figure out how to verify that $ v_t(x,0) = 0$. Here are the steps I have made: \begin{equation*} v_t(x,0) = \partial_{tt} w_\phi(x,0) = \lim_{t\to 0^+} \partial_{tt} w_\phi(x,t) = \lim_{t\to 0^+} c^2\Delta w_\phi(x,t) = c^2\Delta w_\phi(x,0) \end{equation*} where the limits hold due to the regularity of $w_\phi$.

Why is the last term equal to zero? In other words, why is $w_\phi(\cdot,0)$ harmonic?

Thanks in advance.

  • 1
    $\begingroup$ You have specified that $w(x, 0) = 0$. Doesn't this imply $\Delta w_\phi(x, 0) = 0$, since $w_\phi = w$ when $t = 0$? $\endgroup$ – Robert Lewis Sep 3 '18 at 15:45
  • 1
    $\begingroup$ Then why bother using the wave equation in the first place? Since $v(x,0)=\phi(x)$, then $v_t(x,0)=0$ because $\phi$ does not depend on $t$. I have always thought that you should apply the operator first and then calculate the value of the function on the specified point. $\endgroup$ – Giovanni Mariani Sep 3 '18 at 15:51
  • $\begingroup$ Those are two different concepts: the difference between $\lim_{x \rightarrow 0} \frac{d}{dx}(w(x))$ and $\frac{d}{dx}(\lim_{x \rightarrow 0} w(x))$ $\endgroup$ – DaveNine Sep 3 '18 at 18:03
  • $\begingroup$ @DaveNine and that's exactly my point--and my issue: $\Delta w_\phi (x,0) =\lim_{t\to 0^+} \Delta w_\phi (x,t) \neq \Delta \lim_{t\to 0^+} w_\phi (x,t) =\Delta w_\phi (x,0) =\Delta 0 = 0$ $\endgroup$ – Giovanni Mariani Sep 10 '18 at 18:52

I have found a reason why $c^2 \Delta w_\phi(x,0)=0$. Since the Laplacian acts only on the space variables $x$ and leaves the time variable $t$ unchanged, we have $$ (\Delta w_\phi)(x,0) = \Delta (w_\phi(x,0)) = 0 \quad\text{since}\quad w_\phi(x,0)=0 $$ In other words, taking the Laplacian and then letting $t=0$ or vice-versa yield the same result.

Incidentally, this is why one must use the wave equation via the limits and cannot simply stop at $\partial_{tt} w_\phi$: actually $$ (\partial_{tt} w_\phi)(x,0) \neq \partial_{tt} (w_\phi(x,0))$$ My comment above on this point was misleading.


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