# Monge-Ampère Type Function Splittings

## Document Type

Article

## Publication Date

2015

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## Abstract

Given convex u is an element of C((Omega) over bar) with Monge-Ampere measure Mu, and finite Borel measures mu and v satisfying mu + v = Mu, consider the problem of determining a 'splitting' u = v + w for u where v, w is an element of C((Omega) over bar are convex functions satisfying Mv = mu, Mw = v, 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* + w* for u may always be found. In particular, letting U = sup((v, w)) is an element of F (v + w), there exist optimal sums v* + w* achieving inf((v, w)is an element of F) parallel to u - (v + w)parallel to(p) and inf((v, w)is an element of F) parallel to U - (v + w)parallel to(p), p >= 1, for appropriately constrained classes F of feasible pairs (v, w) of convex functionssatisfying Mv = mu, Mw = v and v + w = mu on partial derivative Omega Moreover, U may be written as U = (v) over bar + (w) over bar within (Omega) over bar ,((v) over bar, (w) over bar). 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 is an element of W-2,W- n(Omega) which may be realized as differences u = v - w of convex functions v, w is an element of W-2,W- n(Omega) with Mu = Mv - Mw. Here Mu is the signed measure defined by dMu = det D(2)u dx. Letting U- = sup((v, w)) is an element of F (v - w) and U-= inf((v, w)is an element of F)(v - w), we show that optimal differences v* - w* exist for the problems inf((v, w)) is an element of F parallel to u -(v - w)parallel to(p), inf((v, w))is an element of F parallel to U - (v - w)parallel to(p) and inf((v, w))is an element of F parallel to U- (v -w)parallel to(p). Also, U- = v - w - and U- = v(-) - w(-) for appropriate pairs (v(-), w(-)), (v(-), w(-)) is an element of F. 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.

## Repository Citation

Miller, D.
(2015). Monge-Ampère Type Function Splittings. *Journal of Convex Analysis, 22* (3), 769-796.

https://corescholar.libraries.wright.edu/math/293