claimroutine

Comparison, uniqueness and continuity method setup

Gives comparison, uniqueness and an invertible linearization for continuity
sets up continuity method; supplies uniqueness
Claim·§A The Dirichlet problem on a periodic fiber·inside thmDirichlet problem for quaternionic Monge-Ampère

Statement (paper)

Let ψ1,ψ2\psi_1,\psi_2 be smooth with Qi=H(ψi)>0Q_i=H(\psi_i)>0 on the closure. If det⁡Q1≥det⁡Q2\det Q_1\ge\det Q_2 on the domain and ψ1≤ψ2\psi_1\le\psi_2 on its boundary, then ψ1≤ψ2\psi_1\le\psi_2 throughout; in particular solutions of are unique. Moreover, at each smooth solution ψ=B0+u\psi=B_0+u with Q=H(ψ)>0Q=H(\psi)>0 on the closure, the linearization Lw=Q−1:H(w)Lw=Q^{-1}:H(w) is uniformly elliptic and L:C0k+2,α→Ck,αL:C^{k+2,\alpha}_0\to C^{k,\alpha} is an isomorphism for k≥0k\ge0, 0<α<10<\alpha<1. In the continuity method along gt=(1−t)det⁡H(B0+r)+tgg_t=(1-t)\det H(B_0+r)+tg, starting from u=ru=r at t=0t=0, openness holds by the implicit function theorem, and the C0C^0 estimate follows from the maximum principle with rr as a common strict barrier, so that closedness reduces to the higher-order estimates, with constants uniform in tt.

In plain English

If ψ1,ψ2\psi_1,\psi_2 are smooth with H(ψi)>0H(\psi_i)>0, det⁡H(ψ1)≥det⁡H(ψ2)\det H(\psi_1)\ge\det H(\psi_2) and ψ1≤ψ2\psi_1\le\psi_2 on the boundary, then ψ1≤ψ2\psi_1\le\psi_2 throughout, so solutions of the Dirichlet problem are unique. The linearization Lw=Q−1:H(w)Lw=Q^{-1}:H(w) at a solution is uniformly elliptic and an isomorphism C0k+2,α→Ck,αC^{k+2,\alpha}_0\to C^{k,\alpha}, and the continuity method along gtg_t starting at u=ru=r has open path with a C0C^0 bound, leaving only higher-order estimates for closedness.

Why it's here

This claim supplies the basic tools of the existence proof: the comparison principle gives uniqueness and barrier arguments, and invertibility of the linearization gives openness through the implicit function theorem. It reduces the theorem to a priori estimates uniform in tt, which the later claims prove, starting with the barriers and the reduction of the Hessian bound to the boundary.
Classified routine:
Standard comparison principle and continuity-method setup for Monge–Ampère-type Dirichlet problems (as in Caffarelli–Nirenberg–Spruck for the real case and Alesker for the quaternionic case), adapted here via the paper's Hessian calculus lemma. The argument combines the maximum principle, Schauder theory and the implicit function theorem in the usual way.

Uses

lemHessian calculus in quaternionic coordinates

Used by

claimBarriers, gradient bound, Hessian reduction to boundaryclaimDescent to three slots and parameter dependence

Sources

PaperOpen
Section
§A The Dirichlet problem on a periodic fiber
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sections/dirichlet.tex:48
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