Reference on Minty's trick

Question

I am searching for a precise reference for the following result:

Consider $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ a nondecreasing function.

Assume that a sequence of nonnegative functions $(u_n)_n$ converges weakly in $L^2(\mathbb{R})$ to some element $u$. Assume also that $(f(u_n))_n$ belongs to $L^2(\mathbb{R})$ and converges weakly in this space to some element $v$.

Then, if $(u_n f(u_n))_n$ converges weakly (in the sense of positive measures say) to $uv$, one has $v=f(u)$ a.e.

If furthemore $f$ is strictly increasing, then $(u_n)_n$ converges a.e.

The previous result(s) belong somehow to the folklore of nonlinear PDE and are known with different namings (Minty's trick, Leray-Lions' trick etc).

I am searching for a precise (if possible modern) reference including the strictly increasing case.

Answer 1

I think a standard referenz is "Quelques méthodes de résolution des problèmes aux limites non linéaires" from Lions, but it is in french.

Answer 2

The original reference is:

Minty, George J., on a "monotonicity'' method for the solution of non-linear equations in Banach spaces. Proc. Nat. Acad. Sci. U.S.A. 50 1963 1038–1041. 47.80, https://www.pnas.org/content/50/6/1038

and a "modern" reference, from which your version follows very easily is Lemma D.10 in

The Gradient Discretisation Method : A framework for the discretisation of linear and nonlinear elliptic and parabolic problems. J. Droniou, R. Eymard, T. Gallouët, C. Guichard, R. Herbin. Springer International Publishing AG, série Mathématiques et Applications, (501 pages) - 2018 (online version is "official", and it's in English :)