
Journal of Convex Analysis 22 (2015), No. 3, 769796 Copyright Heldermann Verlag 2015 MongeAmpère Type Function Splittings David F. Miller Dept. of Mathematics and Statistics, Wright State University, Dayton, OH 45435, U.S.A. david.miller@wright.edu [Abstractpdf] Given convex $u\in C(\bar{\Omega})$ with MongeAmp\`{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 $u=vw$ of convex functions $v,w\in W^{2,n}(\Omega)$ with $Mu=MvMw$. Here $Mu$ is the signed measure defined by $dMu={\rm det}~D^2u\,dx$. Letting $U^={\rm sup}_{(v,w)\in {\cal F}}(vw)$ and $U_={\rm inf}_{(v,w)\in {\cal F}}(vw)$, we show that optimal differences $v^\astw^\ast$ exist for the problems ${\rm inf}_{(v,w)\in {\cal F}}~\u(vw)\_p$, ${\rm inf}_{(v,w)\in {\cal F}}~\U^(vw)\_p$ and ${\rm inf}_{(v,w)\in {\cal F}}~\U_ (vw)\_p$. 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. Keywords: MongeAmpere equations, additive solution, optimization characterizations, convex functions. MSC: 20M05; 03D40 [ Fulltextpdf (248 KB)] for subscribers only. 