Manuscript Psy02-B@2 · Psy02-B-v2.tex · theorem · cfd06e2e

Theorem: Energy and geometry stability

Track A4: Gaussian mass energies and stability · lines 910–946 · proof included or linked

Psy02-B@2#thm:massstability

For admissible $f, h$ ,

\abs E_{G} [f + h] - E_{G} [f] \leq G \norm f_{C} \norm h_{C} + G \norm h_{C}^{2} /2.

For Gaussian profiles with widths at least $a_{m i n} > 0$ ,

\norm g_{a}_{C}^{2} = \frac{1}{π a}, \norm \partial_{e} g_{a}_{C}^{2} = \frac{1}{6 π a ^{3}}, \norm \partial_{a} g_{a}_{C}^{2} = \frac{3}{4 π a ^{3}}, \abs e = 1.

Writing

c_{x} = (6 π a_{m i n}^{3})^{- 1/2}, c_{a} = (3/ (4 π a_{m i n}^{3}))^{1/2},

fixed masses and patterns satisfy

\norm δ μ^{'} - δ μ_{C} \leq i \sum m_{i} [c_{x} (2 \abs δ R_{i} + \abs s_{i} δ u) + 2 c_{a} \abs δ a] .

Proof. Expand the positive quadratic form and use Cauchy–Schwarz. The Gaussian derivative norms follow from radial Fourier integrals; rotational averaging contributes a factor $1/3$ for a directional derivative. Integrate translation and width derivatives along their parameter segments and sum the two branch changes.

Displayed formulas in this passage

unlabeled · line 913
$\abs E_{G} [f + h] - E_{G} [f] \leq G \norm f_{C} \norm h_{C} + G \norm h_{C}^{2} /2.$
unlabeled · line 919
$\norm g_{a}_{C}^{2} = \frac{1}{π a}, \norm \partial_{e} g_{a}_{C}^{2} = \frac{1}{6 π a ^{3}}, \norm \partial_{a} g_{a}_{C}^{2} = \frac{3}{4 π a ^{3}}, \abs e = 1.$
unlabeled · line 926
$c_{x} = (6 π a_{m i n}^{3})^{- 1/2}, c_{a} = (3/ (4 π a_{m i n}^{3}))^{1/2},$
unlabeled · line 931
$\norm δ μ^{'} - δ μ_{C} \leq i \sum m_{i} [c_{x} (2 \abs δ R_{i} + \abs s_{i} δ u) + 2 c_{a} \abs δ a] .$

Theorem: Energy and geometry stability

Displayed formulas in this passage

Nearby in this section