The Monge–Ampère equation, stemming from classical geometry, has found significant applications across various fields, including differential geometry, optimal transportation and even image processing. The solvability criterion for systems arising from Monge–Ampère equations plays a pivotal role in understanding the existence and uniqueness of solutions to these equations. In this article, we delve into the mathematical intricacies of this criterion, exploring its theoretical foundations and practical implications. We begin by providing an overview of the Monge–Ampère equation and its significance. Subsequently, we delve into the formulation of systems derived from this equation and elucidate the solvability criterion, discussing its mathematical underpinnings and implications in diverse contexts. Through concrete examples and applications, we illustrate the relevance and utility of this criterion in various mathematical and applied domains.
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