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I am referring to the following version of the theorem, in the setting of the Lebesgue integral. Theorem Let $f: [a,b] \rightarrow \bf R$ be an everywhere differentiable function whose derivative is ...

I am trying to give meaning to the notion of second differential of total variation. For sufficiently regular $u:\Omega \subset \mathbb{R}^2 \to \mathbb{R}$ let the total variation be given by $$TV(u)=...

Consider the following integral equation: $$\int_0^\infty K(t,y)\phi(t,x)dt=0$$ Here, $K(t,y)$ is a trigonometric kernel and $\phi(t,x)$ is monotonic wrt $x$ ( for fixed $t$). I want to find the ...

The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure $\mu$ we have $$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$ for at least one $c\in [a,b]$. We ...

Let $\Omega$ be a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$ and $u:[0,\infty)\times \Omega\to \mathbb{R}$ be a smooth function (for example a smooth solution to a PDE). Thus the ...