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

Two derivatives

Boundary calculations for the finite conversions › Two derivatives · lines 1550–1596

Two integrations by parts give

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

Taylor expansion gives the boundary expression

- σ_{α} ε^{- 2} δ_{ij} f (0) - \frac{3 σ _{α}}{4} \partial_{i} \partial_{j} f (0) - \frac{σ _{α}}{8} δ_{ij} Δ f (0) + O (ε^{2} p_{4} (f)) .

The power and logarithmic terms are exactly those in the direct finite part of $\partial_{i} \partial_{j} u$ . Hence

\partial_{i} \partial_{j} U_{L} = R_{L} [\partial_{i} \partial_{j} u] - \frac{3 σ _{α}}{4} \partial_{i} \partial_{j} δ_{W} - \frac{σ _{α}}{8} δ_{ij} Δ δ_{W} .

Taking a trace yields

Δ U_{L} = 4 V_{L}^{c} - σ_{α} Δ δ_{W} .

Subtracting this trace from the preceding tensor identity and dividing by two gives

\frac{1}{2} K_{ij} U_{L} = S_{L, ij}^{c} + σ_{α} (\frac{7}{16} δ_{ij} Δ - \frac{3}{8} \partial_{i} \partial_{j}) δ_{W} .

This verifies every coefficient in Proposition prop:conversion without numerical execution.

Displayed formulas in this passage

unlabeled · line 1553
$α \int_{r > ε} u \partial_{i} \partial_{j} f d^{2} x - α \int_{r > ε} \partial_{i} \partial_{j} u f d^{2} x = - α {ε^{- 1} \int ω_{i} \partial_{j} f (ε ω) d φ + 2 ε^{- 2} \int ω_{i} ω_{j} f (ε ω) d φ} .$
unlabeled · line 1565
$- σ_{α} ε^{- 2} δ_{ij} f (0) - \frac{3 σ _{α}}{4} \partial_{i} \partial_{j} f (0) - \frac{σ _{α}}{8} δ_{ij} Δ f (0) + O (ε^{2} p_{4} (f)) .$
unlabeled · line 1573
$\partial_{i} \partial_{j} U_{L} = R_{L} [\partial_{i} \partial_{j} u] - \frac{3 σ _{α}}{4} \partial_{i} \partial_{j} δ_{W} - \frac{σ _{α}}{8} δ_{ij} Δ δ_{W} .$
unlabeled · line 1580
$Δ U_{L} = 4 V_{L}^{c} - σ_{α} Δ δ_{W} .$
unlabeled · line 1585
$\frac{1}{2} K_{ij} U_{L} = S_{L, ij}^{c} + σ_{α} (\frac{7}{16} δ_{ij} Δ - \frac{3}{8} \partial_{i} \partial_{j}) δ_{W} .$