claimroutine

Radial solutions dominate log-affine interpolants

Radial solutions of u′′+u′/s=(V+κ2)uu''+u'/s=(V+\kappa^2)u dominate log-affine interpolants
comparison function for outward and return passages
Claim·§6 Propagation across a thin cylinder·inside lemPropagation across a thin cylinder

Statement (paper)

Let S∈{1,3}S\in\{1,3\} and κ>0\kappa>0 be fixed, and let uu solve u′′+s−1u′=(V+κ2)uu''+s^{-1}u'=(V+\kappa^2)u on b<s<Sb<s<S with endpoint values 0,10,1 in either order. Then uu is bounded below by a constant (depending on MM, κ\kappa) times the function affine in log⁡s\log s with the same endpoint values; in the notation x=log⁡(s/b)x=\log(s/b), A=log⁡(S/b)A=\log(S/b), the solution ψ\psi of uxx=P(x)uu_{xx}=P(x)u with ψ(0)=0\psi(0)=0, ψ′(0)=1\psi'(0)=1 satisfies ψ(x)/ψ(A)≥c x/A\psi(x)/\psi(A)\ge c\,x/A on [0,A][0,A], and the same bound holds for the reversed ordering of boundary values.

In plain English

For the radial equation u′′+s−1u′=(V+κ2)uu''+s^{-1}u'=(V+\kappa^2)u on b<s<Sb<s<S with S∈{1,3}S\in\{1,3\} and boundary values 00 and 11 in either order, the solution is at least a constant (depending on M,κM,\kappa) times the interpolant that is affine in log⁡s\log s. In the variable x=log⁡(s/b)x=\log(s/b) this says ψ(x)/ψ(A)≥c x/A\psi(x)/\psi(A)\ge c\,x/A on [0,A][0,A], where ψ′′=Pψ\psi''=P\psi, ψ(0)=0\psi(0)=0, ψ′(0)=1\psi'(0)=1, A=log⁡(S/b)A=\log(S/b), and likewise for the reversed boundary values.

Why it's here

This gives a quantitative lower bound for radial solutions across the thin cylinder, with the loss only a constant multiple of the log-linear profile and no extra powers of DD. The later claims use it as a comparison function to bound the band mass m(s)m(s) from below on the outward passage and on the return passage to inner radius rr, which produces the log⁡(r/b)/D2\log(r/b)/D^2 factor in the lemma.
Classified routine:
A standard ODE comparison argument: after the substitution x=log⁡(s/b)x=\log(s/b) the radial equation becomes ψ′′=Pψ\psi''=P\psi with P≥0P\ge0, and convexity/Sturm comparison bounds the solution below by a constant times the linear interpolant. It is a technical step inside the cylinder lemma, not a new idea.

Uses

Nothing else in the paper

Used by

claimOutward passage bounds band massclaimReturn 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=∣τ∣s=|\tau| drawn schematically. The boundary mass on 1≤T≤21\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[\log(3/b)]^{-2} at the dot. In the wave estimate, b=2Khb=2Kh: HH 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)(T,s) plane, with s=∣τ∣s=|\tau| drawn schematically. Panel (a), "Outward comparison", spans 0≤T≤30\le T\le3 and b≤s≤3b\le s\le3. A thick orange segment labelled "observed wave" lies on the bottom wall s=bs=b for 1≤T≤21\le T\le2, and a shaded band 3/4≤s≤5/43/4\le s\le5/4 is centred on a dashed line at s=1s=1; below it is the factor at s=1s=1, log⁡3/log⁡(3/b)\log 3/\log(3/b). Panel (b), "Inward comparison", spans −1≤T≤1-1\le T\le1 and b≤s≤1b\le s\le1. Its thick teal top wall at s=1s=1 is labelled "uniform lower bound", and dashed guides mark an orange dot at (0,2b)(0,2b) labelled H(0,2b)H(0,2b); below it is the factor at s=2bs=2b, log⁡2/log⁡(1/b)\log 2/\log(1/b). Together the panels show the two logarithmic losses incurred in carrying the energy out to s=1s=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

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Section
§6 Propagation across a thin cylinder
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