claimroutineinferred_dep

Barriers, gradient bound, Hessian reduction to boundary

Gives barriers br≤u≤b′rbr\le u\le b'r and a gradient bound; reduces Hessian to boundary
first a priori estimate for closedness
Claim·§A The Dirichlet problem on a periodic fiber·inside thmDirichlet problem for quaternionic Monge-Ampère

Statement (paper)

There are constants 0<b′<1<b0<b'<1<b, uniform in tt, such that det⁡H(B0+b′r)<gt<det⁡H(B0+br)\det H(B_0+b'r)<g_t<\det H(B_0+br) everywhere, and comparison gives br≤u≤b′rbr\le u\le b'r for the solutions uu along the path. Writing u=αru=\alpha r in a collar with smooth quotient α\alpha, one has b′≤α≤bb'\le\alpha\le b, so ∇u=α∇r\nabla u=\alpha\nabla r on the boundary. Moreover Lr≥cSLr\ge cS and S≥mgt−1/m≥c0>0S\ge m g_t^{-1/m}\ge c_0>0, and for every constant real unit vector ee, Lψe=(log⁡gt)eL\psi_e=(\log g_t)_e and Lψee≥(log⁡gt)eeL\psi_{ee}\ge(\log g_t)_{ee}; consequently the maximum principle applied to ±ψe+Cr\pm\psi_e+Cr bounds the full gradient, and applied to ψee+Cr\psi_{ee}+Cr it reduces the upper bound for the real Hessian to the boundary.

In plain English

Along the continuity path, there are constants 0<b′<1<b0<b'<1<b uniform in tt with br≤u≤b′rbr\le u\le b'r, so u=αru=\alpha r with b′≤α≤bb'\le\alpha\le b and ∇u=α∇r\nabla u=\alpha\nabla r on the boundary. The maximum principle then bounds the full gradient of uu, and reduces the upper bound on the real Hessian to a bound on the boundary.

Why it's here

This is the first a priori estimate for closedness of the continuity method. The functions brbr and b′rb'r serve as sub- and supersolution barriers, and the comparison principle from the previous claim turns them into the C0C^0 bound and a boundary gradient bound. Because Lψe=(log⁡gt)eL\psi_e=(\log g_t)_e and Lψee≥(log⁡gt)eeL\psi_{ee}\ge(\log g_t)_{ee}, with Lr≥cSLr\ge cS and S≥c0>0S\ge c_0>0, adding CrCr gives the gradient bound and leaves only the Hessian on the boundary, which the next claims treat, starting with the mixed derivatives.
Classified routine:
This is the standard Caffarelli–Kohn–Nirenberg–Spruck barrier and maximum-principle argument for complex Monge–Ampère, transferred to the quaternionic setting. It uses sub/supersolutions to get C^0 and boundary gradient bounds, differentiates the equation to bound the gradient, and uses concavity of log det to reduce Hessian bounds to the boundary.

Uses

claimComparison, uniqueness and continuity method setup

Used by

claimMixed normal-tangential derivatives bounded at boundary

Sources

PaperOpen
Section
§A The Dirichlet problem on a periodic fiber
Page
—
Label
—
Line
sections/dirichlet.tex:92
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Barriers, gradient bound, Hessian reduced to boundary" rests on claim-thm-dirichlet-1