Monge-Ampère Type Function Splittings

Given convex $u\in C(\bar{\Omega})$ with Monge-Amp\`{e}re measure $Mu$, and finite Borel measures $\mu$ and $\nu$ satisfying $\mu + \nu = Mu$, consider the problem of determing a `splitting' $u=v+w$ for $u$ where $v,w \in C(\bar{\Omega})$ are convex functions satisfying $Mv=\mu$, $Mw=\nu$, so that $Mu=M(v+w)=Mv + Mw$. It is shown that although this problem is not in general solvable, a best $L^p$ approximation $v^\ast+w^\ast$ for $u$ may always be found. In particular, letting $U={\rm sup}_{(v,w)\in {\cal F}}~(v+w)$, there exist optimal sums $v^\ast+w^\ast$ achieving ${\rm inf}_{(v,w)\in {\cal F}}~\|u-(v+w)\|_p$ and ${\rm inf}_{(v,w)\in {\cal F}}~\|U-(v+w)\|_p$, $p\ge 1$, for appropriately constrained classes ${\cal F}$ of feasible pairs $(v,w)$ of convex functions satisfying $Mv=\mu$, $Mw=\nu$ and $v+w=u$ on $\partial\Omega$. Moreover, $U$ may be written as $U=\bar{v}+\bar{w}$ within $\bar{\Omega}$, $(\bar{v},\bar{w})\in {\cal F}$. The analysis depends upon basic properties of convex functions and the measures they determine. \par\medskip We also consider the related problem of characterizing functions $u\in W^{2,n}(\Omega)$ which may be realized as differences $u=v-w$ of convex functions $v,w\in W^{2,n}(\Omega)$ with $Mu=Mv-Mw$. Here $Mu$ is the signed measure defined by $dMu={\rm det}~D^2u\,dx$. Letting $U^-={\rm sup}_{(v,w)\in {\cal F}}(v-w)$ and $U_-={\rm inf}_{(v,w)\in {\cal F}}(v-w)$, we show that optimal differences $v^\ast-w^\ast$ exist for the problems ${\rm inf}_{(v,w)\in {\cal F}}~\|u-(v-w)\|_p$, ${\rm inf}_{(v,w)\in {\cal F}}~\|U^--(v-w)\|_p$ and ${\rm inf}_{(v,w)\in {\cal F}}~\|U_- -(v-w)\|_p$. Also, $U^-=v^--w^-$ and $U_-=v_--w_-$ for appropriate pairs $(v^-,w^-),(v_-,w_-)\in {\cal F}$. \par\medskip Finally, the relaxed problem of finding $v+w=u$ for general $Mv$ and $Mw$ with $Mv+Mw = Mu$ (no fixed $\mu$ and $\nu$), is considered. Topological properties of the collection of these relaxed splitting pairs $(v,w)$, and those for the unrelaxed problem, for a given $u$, are developed.