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Given are $f\in L^1(\mathbb R^n)$, $f>0$, such that $\log f\in L^1_{\mathrm{loc}}(\mathbb R^n)$ and $\nabla \log f = g$ in the sense of distributions, with $g\in L^1_{\mathrm{loc}}(\mathbb R^n)\cap L^ …

The following paradox has got me stumped. I'm hoping someone can point out the error. Take a locally compact metric space $X$ and define the $C_b(X)$ and $C_0(X)$ as the spaces of continuous real-val …

This paper by Felix Otto applies the method to quasilinear parabolic equations, and explains the steps in detail.

The following theorem seems to have folk status: The topological dual of the space $C_b(X)$ of bounded continuous functions on a topological space $X$ is isomorphic to the space $rba(X)$ of finite, r …

I agree with your edited question. In fact, the information about $u_n$ seems to be unnecessary. Consider $g_n := f_n(u_n)$; then your assumptions give $g_n\rightharpoonup w$ in $L^2(0,T;L^2)$ …