The memoir class provides three-column footnotes. In all it provides four kinds of footnote layouts: normal, two-column, three-column, and run together in a single paragraph.
\documentclass{memoir}
\threecolumnfootnotes
\begin{document}
\null\vfill% just for the example
Some text.\footnote{Author 2001} Some more text.\footnote{Buthor 2002}
And some more.\footnote{Cuthor 2003} Does this text ever end?\footnote{Duthor 2004}
Yes.\footnote{Euthor 2005}
\end{document}
Your problem can be solved with “widetext.sty”. This package mimicing the mechanism with the same name from RevTeX4 was written by Anjishnu Sarkar. Alas, the original home went away. Luckily, we can get a copy from a svn site of Einstein Toolkit, because they have used it for a document (direct link): https://svn.einsteintoolkit.org/documents/Paper_EinsteinToolkit_2010/widetext.sty
Save this package file and put it into your folder for your actual document or, in my eyes better: put into a local TEXMF directory. Since you mentioned MiKTeX in a comment, see Create a local texmf tree in MiKTeX. But if your later want to give away the sources, do not forget to add “widetext.sty”, then.
Here’s an MWE, some explication follows below:
\documentclass[twocolumn]{scrbook}
\usepackage[nohints]{minitoc}
\usepackage{amsmath, amsthm, amssymb, amsfonts, amsbsy}
\usepackage{bigints}
\usepackage{mathtools}
\usepackage{lipsum,kantlipsum}
\usepackage{widetext}% needs packages "flushend" & "cuted" of "sttools"
% bundle, which perhaps must separately be installed
\newcommand{\dd}[1]{\hspace{2pt}d#1}
\begin{document}
\chapter{One}
\lipsum[1]
\begin{widetext}
\begin{equation}\label{fyz:fey_eq_elstat18}
E_x(x_1, y_1, z_1) =
\int\limits_{\substack{\text{all}\\\text{area}}}\varrho(x_2, y_2, z_2)
\frac{x_1-x_2}{[(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2]^{\frac{3}{2}}}\dd{x_2}\dd{y_2}\dd{z_2}
\end{equation}
\end{widetext}
\kant[1]
\end{document}
I intentionally added kantlipsum, another dummy text package, which produces English text. So you can see, how the columns get broken.
There still seem to exist some issues, though, especially with footnotes and floats, coming from the underlying package cuted. Its documentation says, for footnotes you should use \footnotemark plus \footnotetext, but in tests I was not successfull – the notes were printed much to low inside or on top of the right column below the equation depending on the actual text length, cf. code below (I added only modified parts). With \leavevmode\begin{widetext} the output is not perfect, but much better.
Also see on TeX.SE Problems using widetext.sty (do not overlook comments to answers).
\chapter{One}
Text\footnotemark{}
\lipsum[1]
Text\footnotemark
\addtocounter{footnote}{-1}
\footnotetext{Note \thefootnote}
\stepcounter{footnote}
\footnotetext{Note \thefootnote}
\begin{widetext}
...
\end{widetext}
Text\footnote{Note \thefootnote}
\kant[1]
Text\footnote{Note \thefootnote}
Best Answer
I do not think this can be done with LaTeX 2e without Deep Meddling and possibly not at all.
The package documentation is pretty clear: the package involves rewriting internal LaTeX routines controlling output (page 2 section 2.2). This is why it requires two columns as a class feature or option (footnote 7 page 2). It therefore seems unlikely that you can mix the rewritten routines with regular routines in a single document without significant re-engineering, although LaTeX 3 promises greater flexibility (page 2 section 2.2).
If you are prepared to risk Deep Meddling - something which cannot be recommended - then you could - if you really insisted and genuinely pay no heed to the consequences - attempt something along the following sorely misguided lines.
Just remember that when your document breaks, you get to keep every one of the teeny-weeny itsty-bitsy pieces.
Caveat emptor...
Note that I set
\@setmarksto\relaxto allow compilation. I have no idea what this does so this is obviously an extremely foolish thing to do. Did I mention that this approach cannot possibly be recommended?The result produces footnotes which look like this in the first part of the document:
and like this in the second:
which is significantly better than I expected, frankly.