Abstract

Given convex u∈C(Ωˉ)u\in C(\bar{\Omega}) with Monge-Ampère measure MuMu, and finite Borel measures μ\mu and ν\nu satisfying μ+ν=Mu\mu + \nu = Mu, consider the problem of determining a `splitting' u=v+wu=v+w for uu where v,w∈C(Ωˉ)v,w \in C(\bar{\Omega}) are convex functions satisfying Mv=μMv=\mu, Mw=νMw=\nu, so that Mu=M(v+w)=Mv+MwMu=M(v+w)=Mv + Mw. It is shown that although this problem is not in general solvable, a best LpL^p approximation v∗+w∗v^\ast+w^\ast for uu may always be found. In particular, letting U=sup(v,w)∈F (v+w)U={\rm sup}_{(v,w)\in {\cal F}}~(v+w), there exist optimal sums v∗+w∗v^\ast+w^\ast achieving inf(v,w)∈F ∥u−(v+w)∥p{\rm inf}_{(v,w)\in {\cal F}}~\|u-(v+w)\|_p and inf(v,w)∈F ∥U−(v+w)∥p{\rm inf}_{(v,w)\in {\cal F}}~\|U-(v+w)\|_p, p≥1p\ge 1, for appropriately constrained classes F{\cal F} of feasible pairs (v,w)(v,w) of convex functions satisfying Mv=μMv=\mu, Mw=νMw=\nu and v+w=uv+w=u on ∂Ω\partial\Omega. Moreover, UU may be written as U=vˉ+wˉU=\bar{v}+\bar{w} within Ωˉ\bar{\Omega}, (vˉ,wˉ)∈F(\bar{v},\bar{w})\in {\cal F}. The analysis depends upon basic properties of convex functions and the measures they determine. We also consider the related problem of characterizing functions u∈W2,n(Ω)u\in W^{2,n}(\Omega) which may be realized as differences u=v−wu=v-w of convex functions v,w∈W2,n(Ω)v,w\in W^{2,n}(\Omega) with Mu=Mv−MwMu=Mv-Mw. Here MuMu is the signed measure defined by dMu=det D2u dxdMu={\rm det}~D^2u\,dx. Letting U−=sup(v,w)∈F(v−w)U^-={\rm sup}_{(v,w)\in {\cal F}}(v-w) and U−=inf(v,w)∈F(v−w)U_-={\rm inf}_{(v,w)\in {\cal F}}(v-w), we show that optimal differences v∗−w∗v^\ast-w^\ast exist for the problems inf(v,w)∈F ∥u−(v−w)∥p{\rm inf}_{(v,w)\in {\cal F}}~\|u-(v-w)\|_p, inf(v,w)∈F ∥U−−(v−w)∥p{\rm inf}_{(v,w)\in {\cal F}}~\|U^--(v-w)\|_p and inf(v,w)∈F ∥U−−(v−w)∥p{\rm inf}_{(v,w)\in {\cal F}}~\|U_- -(v-w)\|_p. Also, U−=v−−w−U^-=v^--w^- and U−=v−−w−U_-=v_--w_- for appropriate pairs (v−,w−),(v−,w−)∈F(v^-,w^-),(v_-,w_-)\in {\cal F}. Finally, the relaxed problem of finding v+w=uv+w=u for general MvMv and MwMw with Mv+Mw=MuMv+Mw = Mu (no fixed μ\mu and ν\nu), is considered. Topological properties of the collection of these relaxed splitting pairs (v,w)(v,w), and those for the unrelaxed problem, for a given uu, are developed.

Author information

Contact details are reproduced from the original publication and may be historical.

David F. Miller

Dept. of Mathematics and Statistics, Wright State University, Dayton, OH 45435, U.S.A.

david.miller@wright.edu

Keywords

  • Monge-Ampère equations
  • additive solution
  • optimization characterizations
  • convex functions

Mathematics Subject Classification

20M05; 03D40

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Suggested citation

D. F. Miller. “Monge-Ampère Type Function Splittings.” Journal of Convex Analysis 22 (2015), No. 3, 769–796. https://doi.org/10.68381/jca22039

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.