Ytukyg

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 $A$:

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

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

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

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

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

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
$$
-frac{1}{c^{2}}square=left(Delta - frac{1}{c^2}frac{partial^2}{partial t^2}right)chi=f;text{ in };Bbb R^4equiv Bbb R^3times Bbb R label{w}tag{W}
$$
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:
   $$
   square mathscr{E}(x,t)=-c^2delta(x,t)label{da}tag{DA}
   $$
   where $delta(x,t)equiv delta(x)timesdelta(t)$ 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 [1] §8.1-§8.2, pp. 113-116 or [2], §5.1-§5.2, pp. 74-78) by applying to PDE eqref{da} the Fourier transform $mathscr{F}_{xtoxi}$ respect to the spatial variable $x$. By proceeding this way, eqref{da} is transformed into the following ODE:
$$
frac{partial^2 hat{mathscr{E}}(xi,t)}{partial t^2} + c^2|xi|^2hat{mathscr{E}}(xi,t)=-c^2delta(t)label{1}tag{1}
$$
Consider its equivalent standard form
$$
frac{partial^2 hat{mathscr{E}}_p(xi,t)}{partial t^2} + c^2|xi|^2hat{mathscr{E}}_p(xi,t)=delta(t)label{1'}tag{1'}
$$
which has the same solutions, just multiplied by the constant $-c^2$: by solving it (see here, [1] §10.5, p. 147 or [2], §4.9, example 4.9.6 pp. 77-74 and §15.4, example 15.4.4) we get the following distribution
$$
hat{mathscr{E}}_p(xi,t)= H(t)frac{sin c|xi|t}{c|xi|}iffhat{mathscr{E}}(xi,t)= -cH(t)frac{sin c|xi|t}{|xi|}label{2}tag{2}
$$
where $H(t)$ is the usual Heaviside function. Then, taking the inverse Fourier transform $mathscr{F}_{xito x}^{-1}big(hat{mathscr{E}}big)$ we get the sought for solution of eqref{da} (see [1] §9.8, p. 135 and §10.7, p. 149)
$$
mathscr{E}(x,t)=-frac{H(t)}{4pi t}delta_{S_{ct}}(x)=-cfrac{H(t)}{2pi }deltabig(c^2t^2-|x|^2big)label{3}tag{3}
$$
where

• 
  $S_{ct}={xinBbb R^3 | |x|^2=x_1^2+x_2^2+x_3^2=c^2t^2}$ 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 $finmathscr{D}(Bbb R^{3+1})$ 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 $varphiinmathscr{S}$ we have that
$$
begin{split}
langlemathscr{E},varphirangle&=-frac{1}{4pi}intlimits_{0}^{+infty}langledelta_{S_{ct}},varphiranglefrac{mathrm{d}t}{t}\
&=-frac{1}{4pi}intlimits_{0}^{+infty}frac{1}{t}intlimits_{S_{ct}}varphi(x,t),mathrm{d}sigma_xmathrm{d}t
end{split}label{4}tag{4}
$$
From eqref{4} we see that $mathscr{E}$ acts on $varphiinmathscr{S}$ as a spherical mean respect to the spatial $xin Bbb R^3$ variable and as a weighted time integral mean with weight function $tmapsto {1over t}in L^1_mathrm{loc}$ respect to the time variable $tinBbb R_+$.

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 $tinBbb R_+$, 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 $ttoinfty$ a.e. on $Bbb R^3$ with $0<clevarepsilon$.

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.
$$
mathscr{E}astvarphi (x,t) triangleq langle mathscr{E}, varphi(x-y,t-tau)rangle
$$
we get the sought for solution
$$
chi(x,t)=mathscr{E}ast f(x,t)=-frac{1}{4pi}intlimits_{0}^{+infty}frac{1}{tau}intlimits_{S_{ctau}}f(x-y,t-tau),mathrm{d}sigma_ymathrm{d}tau
label{S}tag{WS}
$$

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.

[1] V. S. Vladimirov (1971)[1967], 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.

[2] 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.