Proposition 7.13.3`C006456D2AFC`

Proposition 7.13.3.C006456D2AFC Let $f,g\in W^{2,1}_{\mathrm{loc}}(\Omega )$ be real, and put $E=\{ f=g\} $, formed with any measurable representatives of $f$ and $g$. Then the real weak first and second derivatives of $f$ and $g$ agree almost everywhere on $E$. In particular,

\[ \mathbf1_ E f_{j\bar k}=\mathbf1_ E g_{j\bar k} \quad \text{almost everywhere, for all }j,k. \]

Proof. Apply lemma 7.13.1 to $f-g\in W^{2,1}_{\mathrm{loc}}(\Omega )$, whose zero set, formed with the chosen representatives, is $E$. The complex derivatives $f_{j\bar k}$ and $g_{j\bar k}$ are fixed linear combinations of the real second derivatives, which gives the second assertion. $\square$

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BibTeX citation

@misc{pome-tag-C006456D2AFC,
  author = {{Pome contributors}},
  title = {Proposition 7.13.3},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/C006456D2AFC/}
}

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Proposition 7.13.3C006456D2AFC

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Proposition 7.13.3`C006456D2AFC`