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Generalized Monge-Amp\`ere functionals and related variational problems

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arxiv 2306.01636 v1 pith:I2ZDP6QC submitted 2023-06-02 math.AP math.DG

Freid Tong , Shing-Tung Yau This is my paper · ORCID
classification math.APmath.DG
keywords functionalsassociatedexistencefamilymonge-ampproblemsprovevariational
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In this paper, we introduce a family of real Monge-Amp\`ere functionals and study their variational properties. We prove a Sobolev type inequality for these functionals and use this to study the existence and uniqueness of some associated Dirichlet problems. In particular, we prove the existence of solutions for a nonlinear eigenvalue problem associated to this family of functionals.

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  1. On a general class of free boundary Monge-Amp\`ere equations

    math.AP 2025-08 conditional novelty 8.0 of 10

    A general existence theorem for free-boundary Monge-Ampere equations with prescribed gradient image, applied to degenerate optimal transport, Monge-Ampere eigenvalues, a hemispherical Minkowski problem, and free-bound...

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