claimroutineinferred_dep

Closedness of solvable parameters and boundary smoothness

Shows solvable parameters form a closed set with solutions smooth to the boundary
closedness half of the continuity method
Claim·§A The Dirichlet problem on a periodic fiber·inside thmDirichlet problem for quaternionic Monge-Ampère

Statement (paper)

Let tjt_j be solvable parameters along the continuity path with tj→t∞t_j\to t_\infty. Then a subsequence of the solutions utju_{t_j} converges in C2,βC^{2,\beta}, 0<β<α0<\beta<\alpha, to a limit ut∞u_{t_\infty} satisfying H(B0+ut∞)>0H(B_0+u_{t_\infty})>0, the equation det⁡H(B0+ut∞)=gt∞\det H(B_0+u_{t_\infty})=g_{t_\infty} and ut∞∣∂Ω=0u_{t_\infty}|_{\partial\Omega}=0, and this limit is smooth up to the boundary. Hence the set of solvable parameters is closed, so the path reaches gg.

In plain English

If parameters tjt_j along the continuity path are solvable and tj→t∞t_j\to t_\infty, then a subsequence of the solutions utju_{t_j} converges in C2,βC^{2,\beta} (0<β<α0<\beta<\alpha) to a solution at t∞t_\infty with H(B0+u)>0H(B_0+u)>0, the right determinant equation and zero boundary values, and this limit is smooth up to the boundary. So the set of solvable parameters is closed and the path reaches gg.

Why it's here

The uniform C2,αC^{2,\alpha} bound from the Evans–Krylov claim gives Arzelà–Ascoli compactness, so limits exist and keep the equation and boundary condition. Higher-regularity bootstrapping then makes the limit smooth up to the boundary. Closedness, combined with openness along the path, lets the continuity method reach the target gg, which gives existence in the theorem and feeds the later descent and parameter-dependence step.
Classified routine:
Standard closedness step of the continuity method: Arzelà–Ascoli compactness from the uniform Evans–Krylov C2,αC^{2,\alpha} bound (previous claim) plus Schauder bootstrapping, as in Caffarelli–Kohn–Nirenberg–Spruck's treatment of the Dirichlet problem for Monge–Ampère.

Uses

claimEvans-Krylov C^{2,alpha} estimate

Used by

claimDescent to three slots and parameter dependence

Sources

PaperOpen
Section
§A The Dirichlet problem on a periodic fiber
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sections/dirichlet.tex:196
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Closedness and smoothness up to boundary" rests on claim-thm-dirichlet-5