lemmaroutine

Propagation across a thin cylinder

Subsolution with mild potential loses only

D^{-2}

across a thin cylinder

propagates barrier mass inward in the periodic cap proof

Lemma·lem:cylinder·§6 Propagation across a thin cylinder

Statement (paper)

Let

0<b<1/4

and

D=\log(3/b)

. Suppose a continuous potential

V:[b,3]\to[0,\infty)

satisfies

\int_b^3 s\log(s/b)V(s)\,ds\le M,\qquad \sup_{1/2\le s\le3/2} V(s)\le M .

Let

E\ge0

be smooth on

\mathbb R_T\times \{\tau\in\mathbb R^2:b\le|\tau|\le3\}

, with

\Delta_{T,\tau}E\le V(|\tau|)E

in the interior. Its circular mean

H(T,s)=\frac1{2\pi}\int_0^{2\pi} E\bigl(T,s(\cos\theta,\sin\theta)\bigr)\,d\theta

satisfies, for

b\le r\le1

H(0,r)\ \ge\ c(M)\frac{\log(r/b)}{D^2}\int_1^2 H(T,b)\,dT .

In particular the factor at

r=2b

is a constant times

D^{-2}

In plain English

E\ge0

satisfies

\Delta_{T,\tau}E\le V(|\tau|)E

\mathbb R_T\times\{b\le|\tau|\le3\}

for a potential

V

obeying the moment bound

\int_b^3 s\log(s/b)V\,ds\le M

and

\sup_{1/2\le s\le3/2}V\le M

, then its circular mean satisfies

H(0,r)\ge c(M)\frac{\log(r/b)}{D^2}\int_1^2 H(T,b)\,dT

for

b\le r\le1

, where

D=\log(3/b)

. At

r=2b

the factor is a constant times

D^{-2}

Why it's here

This lemma shows that a nonnegative subsolution with a mild potential cannot lose more than a factor

D^{-2}

of mass between the thin inner circle of radius

b

and the interior, with the extra

\log(r/b)

factor for smaller

r

. The proof sine-weights in

T

to get a radial inequality, then runs a log-affine comparison outward, applies a Harnack step at radius 1, and runs the comparison back inward, each trip costing about one factor of

D

. It supplies the quantitative propagation estimate used by the periodic cap proposition and the later energy bounds.

Classified routine:

The proof chains standard tools: sine-weighting in T to reduce to a radial ODE inequality, comparison with log-affine (2D radial harmonic) barriers, and a Harnack inequality for subsolutions with bounded potential. It is a quantitative but standard barrier/comparison argument, not a headline idea of the paper.

Uses

Nothing else in the paper

Used by

propPeriodic cap estimate with logarithmic lossclaimEnergy bounds give the torus cap estimate

Inside the proof

claimRadial solutions dominate log-affine interpolantsclaimOutward passage bounds band massclaimHarnack lower bound on unit circleclaimReturn passage to inner radius

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: lem:cylinder
Line: sections/propagation.tex:14

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