# solution to $\square\chi=f$.

For an open set $$U \subseteq \mathbb{R}^4$$, if $$f:U \to \mathbb{R}$$ is a "good" (for example, smooth) function, is there a solution to the following equation?

$$\left( \Delta - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\chi(x, y, z, t)=f(x, y, z, t)$$

Context

I want to transform Maxwell's equations

$$\operatorname{rot}E(x,t)+\frac{\partial B(x, t)}{\partial t}=0$$ $$\operatorname{div}B(x,t)=0$$ $$\operatorname{rot}H(x,t)-\frac{\partial D(x,t)}{\partial t}=i(x,t)$$ $$\operatorname{div}D(x,t)=\rho(x,t)$$

into the following form with the electrical potential $$\phi$$ and the vector potential $$A$$:

$$B(x,t)=\operatorname{rot}A_L(x,t)$$ $$E(x,t)=-\frac{\partial A_L(x,t)}{\partial t} -\operatorname{grad}\phi_L(x,t)$$ $$\square A_L(x,t)=-\mu_0i(x,t)$$ $$\square \phi_L(x,t) = -\frac{1}{\epsilon_0}\rho(x,t)$$ $$\operatorname{div}A_L(x, t)+\frac{1}{c^2}\frac{\partial\phi_L(x,t)}{\partial t}=0$$

In order to do this, we need the existence of a solution to the equation

$$\square\chi = -\left(\operatorname{div}A_0 + \frac{1}{c^2}\frac{\partial \phi_0}{\partial t}\right)$$

where $$A_0$$ and $$\phi_0$$ is a special solution to the following equations:

$$B(x,t)=\operatorname{rot}A(x,t)$$ $$E(x,t)=-\frac{\partial A(x,t)}{\partial t} -\operatorname{grad}\phi(x,t)$$ $$\operatorname{grad}\left( \operatorname{div}A(x,t)+\frac{1}{c^2}\frac{\partial \phi(x,t)}{\partial t}\right) + \left( \frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\Delta\right)A(x,t)=\mu_0 i(x,t)$$ $$-\operatorname{div}\left(\frac{\partial A(x,t)}{\partial t}\right) - \Delta \phi(x,t)=\frac{\rho(x,t)}{\epsilon_0}$$

If it exists, $$A_L$$ and $$\phi_L$$ are defined as follows:

$$A_L = A_0 + \operatorname{grad}\chi$$ $$\phi_L = \phi_0 - \frac{\partial}{\partial t}\chi$$

The answer to the question if whether a solution $$\chi$$ to the the following equation exists $$-\frac{1}{c^{2}}\square=\left(\Delta - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\chi=f\;\text{ in }\;\Bbb R^4\equiv \Bbb R^3\times \Bbb R \label{w}\tag{W}$$ under mild smoothness requirements on the datum $$f$$ is yes: I explain below why it is so in a constructive way, by actually constructing an explicit solution in two steps:

1. Construction of a fundamental solution: what is needed is a slightly modified fundamental solution of the D'Alembert operator, precisely the solution of the following equation: $$\square \mathscr{E}(x,t)=-c^2\delta(x,t)\label{da}\tag{DA}$$ where $$\delta(x,t)\equiv \delta(x)\times\delta(t)$$ is the usual tensor product of Dirac measures respectively on the spatial and on the time domain. Once $$\mathscr{E}(x,t)$$ has been determined, we can find, provided certain compatiility conditions on $$f$$ are fulfilled (see below), a distributional solution $$\chi(x,t)$$ to the posed problem by convolution $$\chi(x,t)=\mathscr{E}\ast f(x,t)\label{s}\tag{S}$$ The minimal requirements on $$f$$ is that the convolution product at the right term of \eqref{s} should exists as a distribution.

2. The regularity problem: prove that, provided $$f$$ is a "good" (for example $$C^2$$ smooth) function, the distribution $$\chi$$ in \eqref{s} is a "good" function in the same way.

Calculation of the modified fundamental solution for the D'Alembert operator in $$\Bbb R^{3+1}$$

We construct $$\mathscr{E}$$ as a distribution of slow growth (i.e. $$\mathscr{E}\in \mathscr{S}^\prime$$, see for example  §8.1-§8.2, pp. 113-116 or , §5.1-§5.2, pp. 74-78) by applying to PDE \eqref{da} the Fourier transform $$\mathscr{F}_{x\to\xi}$$ respect to the spatial variable $$x$$. By proceeding this way, \eqref{da} is transformed into the following ODE: $$\frac{\partial^2 \hat{\mathscr{E}}(\xi,t)}{\partial t^2} + c^2|\xi|^2\hat{\mathscr{E}}(\xi,t)=-c^2\delta(t)\label{1}\tag{1}$$ Consider its equivalent standard form $$\frac{\partial^2 \hat{\mathscr{E}}_p(\xi,t)}{\partial t^2} + c^2|\xi|^2\hat{\mathscr{E}}_p(\xi,t)=\delta(t)\label{1'}\tag{1'}$$ which has the same solutions, just multiplied by the constant $$-c^2$$: by solving it (see here,  §10.5, p. 147 or , §4.9, example 4.9.6 pp. 77-74 and §15.4, example 15.4.4) we get the following distribution $$\hat{\mathscr{E}}_p(\xi,t)= H(t)\frac{\sin c|\xi|t}{c|\xi|}\iff\hat{\mathscr{E}}(\xi,t)= -cH(t)\frac{\sin c|\xi|t}{|\xi|}\label{2}\tag{2}$$ where $$H(t)$$ is the usual Heaviside function. Then, taking the inverse Fourier transform $$\mathscr{F}_{\xi\to x}^{-1}\big(\hat{\mathscr{E}}\big)$$ we get the sought for solution of \eqref{da} (see  §9.8, p. 135 and §10.7, p. 149) $$\mathscr{E}(x,t)=-\frac{H(t)}{4\pi t}\delta_{S_{ct}}(x)=-c\frac{H(t)}{2\pi }\delta\big(c^2t^2-|x|^2\big)\label{3}\tag{3}$$ where

• $$S_{ct}=\{x\in\Bbb R^3 | |x|^2=x_1^2+x_2^2+x_3^2=c^2t^2\}$$ is the spherical light wave surface,
• $$\delta_{S_{ct}}(x)$$ is the Dirac measure supported on $$S_{ct}$$, otherwise called single layer measure.

Now, given any distribution $$f\in\mathscr{D}(\Bbb R^{3+1})$$ for which the convolution with $$\mathscr{E}$$ exists (for example any distribution of compact support) using \eqref{3} in formula \eqref{s} gives a generalized solution of \eqref{w}.

Construction of a regular solution

Instead of recurring to the standard (and complex) methods of regularity theory we will try a trickier way by looking carefully at the structure of \eqref{3} and on how this distribution acts on the space of infinitely smooth rapidly decreasing functions: precisely, given $$\varphi\in\mathscr{S}$$ we have that $$\begin{split} \langle\mathscr{E},\varphi\rangle&=-\frac{1}{4\pi}\int\limits_{0}^{+\infty}\langle\delta_{S_{ct}},\varphi\rangle\frac{\mathrm{d}t}{t}\\ &=-\frac{1}{4\pi}\int\limits_{0}^{+\infty}\frac{1}{t}\int\limits_{S_{ct}}\varphi(x,t)\,\mathrm{d}\sigma_x\mathrm{d}t \end{split}\label{4}\tag{4}$$ From \eqref{4} we see that $$\mathscr{E}$$ acts on $$\varphi\in\mathscr{S}$$ as a spherical mean respect to the spatial $$x\in \Bbb R^3$$ variable and as a weighted time integral mean with weight function $$t\mapsto {1\over t}\in L^1_\mathrm{loc}$$ respect to the time variable $$t\in\Bbb R_+$$.
This implies that \eqref{4} is meaningful also for functions which are not in $$\mathscr{S}$$ nor are infinitely smooth. Precisely, provided that

• $$\varphi(\cdot,t)\in L^1_\mathrm{loc}(\Bbb R^3)$$ for almost all $$t\in\Bbb R_+$$, without any growth condition at infinity and
• $$\varphi(x,\cdot)\in L^1_\mathrm{loc}(\Bbb R)$$ with $$|\varphi(x,t)|=O(t^{-\varepsilon})$$ as $$t\to\infty$$ a.e. on $$\Bbb R^3$$ with $$0.

equation \eqref{4} is meaningful. Then, by putting $$\varphi(y,\tau)=f(x-y,t-\tau)$$ and by using \eqref{4} jointly with the definition of convolution between a distribution and a function, i.e. $$\mathscr{E}\ast\varphi (x,t) \triangleq \langle \mathscr{E}, \varphi(x-y,t-\tau)\rangle$$ we get the sought for solution $$\chi(x,t)=\mathscr{E}\ast f(x,t)=-\frac{1}{4\pi}\int\limits_{0}^{+\infty}\frac{1}{\tau}\int\limits_{S_{c\tau}}f(x-y,t-\tau)\,\mathrm{d}\sigma_y\mathrm{d}\tau \label{S}\tag{WS}$$

Notes

• The hypothesis $$n=3$$, i.e. the fact that we are working in a $$3$$D space, is essential for defining the structure of \eqref{3}. The inverse transform of $$\hat{\mathscr{E}}$$ in \eqref{2} has not the same structure on every $$\Bbb R^n$$: in monographs on hyperbolic PDEs, this concept is also stated by saying that Huygens's principle does not hold in even spatial dimension.
• The regularity of the solution we have obtained is very weak: in particularly we do not know the smoothness of $$\chi$$ for a given smoothness of $$f$$. Deeper methods are required for the investigation of this problems.

 V. S. Vladimirov (1971), Equations of mathematical physics, Translated from the Russian original (1967) by Audrey Littlewood. Edited by Alan Jeffrey, (English), Pure and Applied Mathematics, Vol. 3, New York: Marcel Dekker, Inc., pp. vi+418, MR0268497, Zbl 0207.09101.

 V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029.