claimroutineinferred_dep

Mixed normal-tangential derivatives bounded at boundary

Bounds mixed normal-tangential second derivatives at the boundary by

C

boundary Hessian entry needed for double-normal bound

Claim·§A The Dirichlet problem on a periodic fiber·inside thmDirichlet problem for quaternionic Monge-Ampère

Statement (paper)

Let

\nu

be the outward unit normal at a boundary point and let

e

be a real tangent unit vector there. Then, with constants uniform over the compact boundary,

|D^2u(\nu,e)|\le C

, where

C

depends only on the data. That is, the mixed second derivatives of

u

in one normal and one real tangential direction are bounded at the boundary, and the tangential-tangential second derivatives satisfy

D^2u(v,w)=\alpha D^2r(v,w)

In plain English

At a boundary point, with

\nu

the outward unit normal and

e

a real tangent unit vector, the mixed second derivative satisfies

|D^2u(\nu,e)|\le C

with

C

depending only on the data. The tangential-tangential second derivatives are determined by

D^2u(v,w)=\alpha D^2r(v,w)

Why it's here

Since

u=\alpha r

on the boundary with

\nabla u=\alpha\nabla r

, the tangential-tangential Hessian is fixed by

r

, so the mixed normal-tangential entries are the next unknowns in the boundary Hessian. The bound is obtained through a quaternionic rigid-motion argument that uses the covariance of

H

from the Hessian calculus. It is the input for the double normal derivative estimate, which together with this claim gives the full

C^2

bound at the boundary.

Classified routine:

This is the standard mixed normal-tangential boundary C^2 estimate for the Dirichlet problem for Monge-Ampère type equations, in the style of Caffarelli-Kohn-Nirenberg-Spruck. Here it is adapted to the quaternionic Hessian using rigid motions and the covariance of H.

Uses

claimBarriers, gradient bound, Hessian reduction to boundarylemHessian calculus in quaternionic coordinates

Used by

claimDouble normal derivative bound and full C2

Sources

PaperOpen

Section: §A The Dirichlet problem on a periodic fiber
Page: —
Label: —
Line: sections/dirichlet.tex:112

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Mixed boundary derivatives via quaternionic rigid motion" rests on claim-thm-dirichlet-2
info
inferred_dep
outline: "Mixed boundary derivatives via quaternionic rigid motion" rests on lem:hessian-calculus