Newton-Leibniz formula in $H^m(\Omega)$

by whereamI   Last Updated June 20, 2018 04:20 AM

I know that one can prove that for $u\in H^1(-1,1)$, the following equality hods: $$ u(x)=u(-1)+\int^x_{-1}u'(\tilde{x})d\tilde{x},\quad x\in (-1,1) $$

But can we obtain a kind of Newton-Leibniz formula over weak partial derivative in high dimension?

For an example, let $\Omega=[-1,1]\times [-1,1]\subset\mathbb{R}^2$, and $u\in H^2(\Omega)$. By embedding theorem, we have $u\in C(\Omega)$. Can we have following equalities: $$ \begin{aligned} u(x,y)&=\int_{-1}^xu_{x}(\tilde{x},y)d\tilde{x}+u(-1,y)\\ \end{aligned} $$ By Fubini's theorem, one can obtain that $\int_{-1}^xu_{x}(\tilde{x},y)d\tilde{x}$ is finite for almost every $y$. But can we prove the above equality holds?



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