claimroutineinferred_dep

Outward passage bounds band mass

Outward passage gives

m(s)\ge c\,m(b)\log(3/s)/D

for the sine-weighted mean

first

1/D

loss; feeds the Harnack step

Claim·§6 Propagation across a thin cylinder·inside lemPropagation across a thin cylinder

Statement (paper)

Let

m(s)=\int_0^3\sin(\pi T/3)H(T,s)\,dT

. Then

m''+s^{-1}m'\le (V+\pi^2/9)m

, and

m(s)\ge c\,m(b)\log(3/s)/D

for

b\le s\le 3

. In particular,

\int_0^3\int_{3/4\le|\tau|\le5/4}H(T,|\tau|)\,d\tau\,dT\ \ge\ c\,m(b)/D

In plain English

Let

m(s)=\int_0^3\sin(\pi T/3)H(T,s)\,dT

be the sine-weighted mean of

H

T

. Then

m''+s^{-1}m'\le (V+\pi^2/9)m

, and

m(s)\ge c\,m(b)\log(3/s)/D

for

b\le s\le 3

, so the integral of

H(T,|\tau|)

over

0\le T\le 3

and

3/4\le|\tau|\le 5/4

is at least

c\,m(b)/D

Why it's here

Weighting by

\sin(\pi T/3)

and integrating by parts in

T

turns the subsolution inequality for

E

into a radial differential inequality for

m

with the extra term

\pi^2/9

. The comparison claim with the log-affine interpolant then gives

m(s)\gtrsim m(b)\log(3/s)/D

with no loss beyond a constant, which lets the later Harnack step pass from the inner boundary mass

m(b)

H

on the unit circle at cost

1/D

Classified routine:

Obtained by a standard argument: test the subsolution inequality against the first Dirichlet eigenfunction sin(πT/3), integrate by parts in T, then apply the earlier log-affine comparison claim (claim-lem-cylinder-1) through an ODE maximum-principle comparison.

Uses

claimRadial solutions dominate log-affine interpolants

Used by

claimHarnack lower bound on unit circle

Figures

Two side-by-side rectangles in the $(T,s)$ plane, with $s=|\tau|$ drawn schematically. Panel (a), "Outward comparison", spans $0\le T\le3$ and $b\le s\le3$. A thick orange segment labelled "observed wave" lies on the bottom wall $s=b$ for $1\le T\le2$, and a shaded band $3/4\le s\le5/4$ is centred on a dashed line at $s=1$; below it is the factor at $s=1$, $\log 3/\log(3/b)$. Panel (b), "Inward comparison", spans $-1\le T\le1$ and $b\le s\le1$. Its thick teal top wall at $s=1$ is labelled "uniform lower bound", and dashed guides mark an orange dot at $(0,2b)$ labelled $H(0,2b)$; below it is the factor at $s=2b$, $\log 2/\log(1/b)$. Together the panels show the two logarithmic losses incurred in carrying the energy out to $s=1$ and back to the inner wall. — Figure 2schematicp. 22
The two comparisons, with radii $s=|\tau|$ drawn schematically. The boundary mass on $1\le T\le2$ bounds the mass in the shaded band from below. The mean inequality on a fixed annulus supplies a lower bound on the entire upper wall in (b). The two displayed fractions are values of the normalized radial harmonic functions. The comparison solutions with potential dominate fixed multiples of these values, giving the factor $[\log(3/b)]^{-2}$ at the dot. In the wave estimate, $b=2Kh$ : $H$ controls the wave norm on the marked segment and is controlled by the initial data at the dot.
What it shows
Two side-by-side rectangles in the $(T,s)$ plane, with $s=|\tau|$ drawn schematically. Panel (a), "Outward comparison", spans $0\le T\le3$ and $b\le s\le3$ . A thick orange segment labelled "observed wave" lies on the bottom wall $s=b$ for $1\le T\le2$ , and a shaded band $3/4\le s\le5/4$ is centred on a dashed line at $s=1$ ; below it is the factor at $s=1$ , $\log 3/\log(3/b)$ . Panel (b), "Inward comparison", spans $-1\le T\le1$ and $b\le s\le1$ . Its thick teal top wall at $s=1$ is labelled "uniform lower bound", and dashed guides mark an orange dot at $(0,2b)$ labelled $H(0,2b)$ ; below it is the factor at $s=2b$ , $\log 2/\log(1/b)$ . Together the panels show the two logarithmic losses incurred in carrying the energy out to $s=1$ and back to the inner wall.
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SupportslemPropagation across a thin cylinderclaimOutward passage bounds band massclaimHarnack lower bound on unit circleclaimReturn passage to inner radiusclaimRadial solutions dominate log-affine interpolants

Sources

PaperOpen

Section: §6 Propagation across a thin cylinder
Page: —
Label: —
Line: sections/propagation.tex:63

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Outward passage bounds band mass" rests on claim-lem-cylinder-1