theoremroutineunresolved_ref

Dirichlet problem for quaternionic Monge-Ampère

Solves

\det H(B_0+u)=g

with zero boundary values, uniquely, with barriers

supplies the Dirichlet solution for the barrier argument

Theorem·thm:dirichlet·§A The Dirichlet problem on a periodic fiber

Statement (paper)

Let

m\ge1

. Suppose

r\in C^\infty(\overline V)

is a defining function, negative in

V

, with

dr\ne0

\partial V

and

H(r)>0

\overline\Omega

. Let

B_0,g\in C^\infty(\overline\Omega)

be real and suppose

H(B_0)\ge0,\qquad g>\det H(B_0)\quad\text{on }\overline\Omega.

There is a unique smooth solution, up to the boundary, of

\det H(B_0+u)=g,\qquad H(B_0+u)>0,\qquad u|_{\partial\Omega}=0.

There are constants

b\ge b'>0

such that

br\le u\le b'r.

For smooth finite-dimensional families of such data and domains, these constants can be chosen locally uniformly. The solutions depend smoothly on the parameters up to the boundary, after any smooth local identification of the domains with a fixed domain.

In plain English

On a domain with a defining function

r

for which

H(r)>0

, if

H(B_0)\ge0

and

g>\det H(B_0)

, then

\det H(B_0+u)=g

with

H(B_0+u)>0

and

u|_{\partial\Omega}=0

has a unique solution smooth up to the boundary. It satisfies barrier bounds

br\le u\le b'r

with

b\ge b'>0

, and the constants and solution depend smoothly and locally uniformly on parameters of the data and domains.

Why it's here

This is the existence theorem for the Dirichlet problem that later barrier and pencil-fiber arguments rely on. The proof runs a continuity method on a lifted domain, with comparison and uniqueness, barrier and gradient bounds, boundary Hessian bounds using the Hessian calculus and the cofactor/Schur facts, and an Evans–Krylov estimate for closedness. It then descends to the three-slot domain and gets parameter dependence from the implicit function theorem.

Classified routine:

Adapts the known solvability of the Dirichlet problem for quaternionic Monge–Ampère on strictly pseudoconvex domains (Alesker, following Caffarelli–Kohn–Nirenberg–Spruck) to this background-form, lifted/three-slot setting. It uses the standard continuity method with barrier, gradient, boundary Hessian and Evans–Krylov estimates, plus the implicit function theorem for parameter dependence.

Uses

lemHessian calculus in quaternionic coordinateslemMoore determinant and cofactor properties

Used by

propEnergy barrier on a fixed fiberclaimUpper bound on boundary quotient u/rclaimInterior energy lower bound via ball quadraticclaimDeterminant energy and balance on pencil fibers

Inside the proof

claimComparison, uniqueness and continuity method setupclaimBarriers, gradient bound, Hessian reduction to boundaryclaimMixed normal-tangential derivatives bounded at boundaryclaimDouble normal derivative bound and full C2claimEvans-Krylov C^{2,alpha} estimateclaimClosedness of solvable parameters and boundary smoothnessclaimDescent to three slots and parameter dependence

Sources

PaperOpen

Section: §A The Dirichlet problem on a periodic fiber
Page: —
Label: thm:dirichlet
Line: sections/dirichlet.tex:24

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

warn
unresolved_ref
C^{2,alpha} and Schauder steps rest on external Gilbarg-Trudinger, Evans-Krylov and Caffarelli-Nirenberg-Spruck results.