Elliptic theorem of sums with the full block matrix bound

First prove the quadratic compactly localized case. Translate to $z=0$, subtract $u_i(0)$ and the affine terms $D_i\varphi (0)\cdot x_i$, and suppose $u_i(0)=0$ and $w(x):={\sum }_i{u}_i(x_i)\le x^{\mathsf{T}}Ax\mathrm{/}2$ on compact neighborhoods $K_i$ of zero. Extend the restrictions by $-\mathrm{\infty }$ off $K_i$, set $\lambda ={\epsilon }^{-1}+\mathrm{\parallel }A\mathrm{\parallel }>0$ and $B=A+\epsilon A^2$, and write $h=x-\xi$. The identity $\mathrm{\mid }{\epsilon }^{-1\mathrm{/}2}h-{\epsilon }^{1\mathrm{/}2}A\xi {\mathrm{\mid }}^2\ge 0$ gives $x^{\mathsf{T}}Ax\le {\xi }^{\mathsf{T}}B\xi +\lambda \mathrm{\mid }x-\xi {\mathrm{\mid }}^2$. Define ${\widehat{u}}_i$ by [F2] and $f(\xi )={\sum }_i{\widehat{u}}_i({\xi }_i)$. The supremum separates over the finite compact product, so $f(\xi )={\sup }_x\{w(x)-\lambda \mathrm{\mid }x-\xi {\mathrm{\mid }}^2\mathrm{/}2\}\le {\xi }^{\mathsf{T}}B\xi \mathrm{/}2$. Each ${\widehat{u}}_i(0)\ge 0$, while $f(0)\le 0$, so all are zero; also $f+\lambda \mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{/}2$ is convex.

The function $F(\xi )=f(\xi )-{\xi }^{\mathsf{T}}B\xi \mathrm{/}2-\mathrm{\mid }\xi {\mathrm{\mid }}^4$ has a strict maximum at zero and is locally semiconvex, since the Hessian of the subtracted polynomial is bounded on every fixed ball. Apply [F3] outside the null set where $f$ lacks a Peano expansion: for each $n$, [A1] selects $\mathrm{\mid }{\xi }_n\mathrm{\mid }<1\mathrm{/}n$, $\mathrm{\mid }{q}_n\mathrm{\mid }\le 1\mathrm{/}n$ so $F+q_n\cdot \xi$ has a local maximum at ${\xi }_n$ and $f$ has expansion $(p_n,Z_n)$ there. Restricting to lines gives $p_n=B{\xi }_n+4\mathrm{\mid }{\xi }_n{\mathrm{\mid }}^2{\xi }_n-q_n$ and $Z_n\le B+8{\xi }_n{\xi }_n^{\mathsf{T}}+4\mathrm{\mid }{\xi }_n{\mathrm{\mid }}^{2}I$; convexity of $f+\lambda \mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{/}2$ gives $Z_n\ge -\lambda I$. Thus $p_n\to 0$ and $Z_n$ is bounded. AC supplies the countable selections and is inherited from [F3].

Separability makes $Z_n$ block diagonal: each coordinate restriction gives the corresponding Peano expansion of ${\widehat{u}}_i$, and summing those expansions gives a second expansion of $f$ with Hessian $\mathrm{diag}(Z_{n,11},\dots ,Z_{n,kk})$. Uniqueness follows because a quadratic form that is $o(\mathrm{\mid }h{\mathrm{\mid }}^2)$ vanishes on every direction and hence by polarization. Set $X_{n,i}=Z_{n,ii}$. The transfer in [F2] gives $y_{n,i}={\xi }_{n,i}+p_{n,i}\mathrm{/}\lambda$, $(p_{n,i},X_{n,i})\in J^{2,+}{u}_i(y_{n,i})$, and $u_i(y_{n,i})={\widehat{u}}_i({\xi }_{n,i})+\mathrm{\mid }{p}_{n,i}{\mathrm{\mid }}^2\mathrm{/}(2\lambda )$. Since $y_{n,i}\to 0$, they eventually lie inside $K_i$; continuity of ${\widehat{u}}_i$ gives $u_i(y_{n,i})\to 0=u_i(0)$.

By [F4], extract one subsequence along which the finitely many entries of $Z_n$ converge to $Z=\mathrm{diag}(X_i)$. The inequalities in step 2.1 pass to the limit, giving $-\lambda I\le Z\le B$. The convergent jets and values from step 3.1 imply $(0,X_i)\in {\bar{J}}^{2,+}{u}_i(0)$ by [F1]; this proves the localized quadratic case.

For the general case, translate and subtract affine functions as above. Taylor's formula gives $\varphi (x)=x^{\mathsf{T}}Ax\mathrm{/}2+o(\mathrm{\mid }x{\mathrm{\mid }}^2)$, so for every sufficiently small $\eta >0$ there are compact neighborhoods $K_i^{\eta }$ with ${\sum }_i{u}_i(x_i)\le x^{\mathsf{T}}(A+\eta I)x\mathrm{/}2$. Apply the quadratic case from step 4.1 with $A_{\eta }=A+\eta I$; it gives closed jets $(0,X_i^{\eta })$ with $-({\epsilon }^{-1}+\mathrm{\parallel }{A}_{\eta }\mathrm{\parallel })I\le \mathrm{diag}(X_i^{\eta })\le A_{\eta }+\epsilon A_{\eta }^2$. Because zero is interior to each $K_i^{\eta }$, localization does not change these closed jets.

Take ${\eta }_n=1\mathrm{/}n$ and, by [A1], choose for each sufficiently large $n$ one tuple of closed-jet matrices supplied by step 5.1. Use [F4] to extract a convergent subsequence of these block matrices. The limit obeys the required bounds because $\mathrm{\parallel }A+\eta I\mathrm{\parallel }\to \mathrm{\parallel }A\mathrm{\parallel }$. For each closed jet $(0,X_i^{\eta })$, [A1] selects an actual approximating jet with point, value, gradient, and matrix within $1\mathrm{/}n$; the defining value-sensitive closure [F1] then gives the limiting closed jet. Restoring the subtracted affine terms gives gradient $D_i\varphi (z)$, and negating jets proves the lower-semicontinuous version.