3
$\begingroup$

Q1: Where can I find a modern detailed exposition of Kruzkov's doubling of variables method and its applications?

Improve this question
$\endgroup$

2 Answers 2

Reset to default
3
$\begingroup$

Ben Andrews and I used and extended this method in a series of papers: for parabolic equations in one space dimension here, for general dimensions here. We also used it for an application to an eigenvalue problem and also on manifolds (those papers are also on arxiv).

We were more influenced by Huisken's paper on the Grayson result, though in the end what we have looks closer to Kruzkhov. There is also extensive treatment in the stochastic pde literature (the coupling method).

Julie

Improve this answer
$\endgroup$
1
  • $\begingroup$ Hi Julie, would you mind elaborating how the doubling of variables method is related to the coupling method in stochastic PDEs? $\endgroup$
    – Peter Wacken
    Commented Mar 28, 2023 at 18:51
1
$\begingroup$

One standad application is the proof of uniqueness of viscosity solutions. One presentation is in Evans, Partial Differential Equations (2 ed), in the proof of Theorem 1 p. 589. See also S. Koike (Department of Mathematics, Saitama University, Japan), A Beginner’s Guide to the Theory of Viscosity Solutions, 2nd edition (version: June 28, 2012). It can be found here: www.math.tohoku.ac.jp/~koike/evis2012version.pdf

Improve this answer
$\endgroup$

You must log in to answer this question.