claimroutineinferred_dep

Determinant energy and balance on pencil fibers

Bounds ΔF\Delta\mathcal F by a wall integral for the pencil determinant energy
specializes the energy flux machinery to pencils
Claim·§E.3 The fixed cap estimate·inside thmLocal smoothing for symmetric pencil waves

Statement (paper)

With A=Aw+h50nC(H(r))A=A_{\rm w}+h^{50n}C(H(r)), μ=ATT\mu=A_{TT} and B0=λ∣Φ∣2B_0=\lambda|\Phi|^2, λ=h−30n\lambda=h^{-30n}, the conserved positive semidefinite tensor AA satisfies A:H(B0)=0A:H(B_0)=0 and μ>0\mu>0, and the solution u≤0u\le0 of given by Theorem  has energy F=Jz(u)\mathcal F=\mathcal J_z(u) satisfying F≍∫r<0(−u)μ dX  dP\mathcal F\asymp\int_{r<0}(-u)\mu\dd X\,\dd P and ΔT,τF≤∫(u/r)G(r)∗AwG(r) δ(r) dX  dP\Delta_{T,\tau}\mathcal F\le\int (u/r)G(r)^*A_{\rm w}G(r)\,\delta(r)\dd X\,\dd P.

In plain English

With A=Aw+h50nC(H(r))A=A_{\rm w}+h^{50n}C(H(r)), μ=ATT\mu=A_{TT} and B0=λ∣Φ∣2B_0=\lambda|\Phi|^2, λ=h−30n\lambda=h^{-30n}, the conserved positive semidefinite tensor AA satisfies A:H(B0)=0A:H(B_0)=0 and μ>0\mu>0. The Dirichlet solution u≤0u\le0 has fiber energy F=Jz(u)\mathcal F=\mathcal J_z(u) comparable to ∫r<0(−u)μ  dX  dP\int_{r<0}(-u)\mu\,\dd X\,\dd P, and ΔT,τF\Delta_{T,\tau}\mathcal F is bounded above by the wall integral ∫(u/r)G(r)∗AwG(r) δ(r)  dX  dP\int (u/r)G(r)^*A_{\rm w}G(r)\,\delta(r)\,\dd X\,\dd P.

Why it's here

This specializes the abstract energy-flux machinery to the pencil wave. The tensor AA combines the conserved current tensor from the Gaussian-multiplier claim with the divergence-free cofactor C(H(r))C(H(r)), which gives A:H(B0)=0A:H(B_0)=0 and μ>0\mu>0 so that Theorem A.1 yields uu. The fiber energy comparison then makes F\mathcal F comparable to ∫(−u)μ\int(-u)\mu, and the energy-flux identity gives the one-sided Laplacian bound; the later energy bounds and the torus cap estimate use both.
Classified routine:
This node assembles earlier results: the conserved current tensor, the divergence-free cofactor lemma, the Dirichlet theorem, the energy comparison and the energy-flux identity. Specializing them to the pencil wave with these parameter choices is a direct consequence of those nodes, so no new idea is introduced here.

Uses

claimConserved current with Gaussian multiplier symbolsthmDirichlet problem for quaternionic Monge-AmpèrelemHessian cofactor is column-divergence-freepropEnergy flux identity with full wall normallemFiber energy comparable to X_0

Used by

claimEnergy bounds give the torus cap estimate

Sources

PaperOpen
Section
§E.3 The fixed cap estimate
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sections/pencil.tex:190
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Determinant energy and balance on pencil fibers" rests on claim-thm-pencil-3
  • info
    inferred_dep
    outline: "Determinant energy and balance on pencil fibers" rests on thm:dirichlet
  • info
    inferred_dep
    outline: "Determinant energy and balance on pencil fibers" rests on lem:cofactor-divergence
  • info
    inferred_dep
    outline: "Determinant energy and balance on pencil fibers" rests on prop:energy-flux
  • info
    inferred_dep
    outline: "Determinant energy and balance on pencil fibers" rests on lem:energy-comparison