Manuscript I-B-VP-1@1 · I-B-VP-1-v1.tex · section · 6d700be9

One derivative

Boundary calculations for the finite conversions › One derivative · lines 1525–1549

For $u = r^{- 2}$ , integration by parts on $r > ε$ uses the inner outward normal $- ω$ :

α \int_{r > ε} u \partial_{i} f d^{2} x = - α \int_{r > ε} \partial_{i} u f d^{2} x - α ε^{- 1} \int_{0}^{2 π} ω_{i} f (ε ω) d φ .

The last expression equals

- \frac{σ _{α}}{2} \partial_{i} f (0) + O (ε^{2} p_{3} (f)) .

After including the logarithmic subtraction in $U_{L}$ , this gives

\partial_{i} U_{L} = R_{L} [\partial_{i} u] - \frac{σ _{α}}{2} \partial_{i} δ_{W} .

Since $\partial_{i} u = - 2 x_{i} / r^{4}$ , this is (eq:convertQ).

Displayed formulas in this passage

unlabeled · line 1529
$α \int_{r > ε} u \partial_{i} f d^{2} x = - α \int_{r > ε} \partial_{i} u f d^{2} x - α ε^{- 1} \int_{0}^{2 π} ω_{i} f (ε ω) d φ .$
unlabeled · line 1537
$- \frac{σ _{α}}{2} \partial_{i} f (0) + O (ε^{2} p_{3} (f)) .$
unlabeled · line 1542
$\partial_{i} U_{L} = R_{L} [\partial_{i} u] - \frac{σ _{α}}{2} \partial_{i} δ_{W} .$