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.

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.

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.

Any suggestions ?

Thanks !

Ayman

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.