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-boundary toric Kaehler-Ricci solitons.
Generalized Monge-Amp\`ere functionals and related variational problems
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abstract
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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On a general class of free boundary Monge-Amp\`ere equations
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-boundary toric Kaehler-Ricci solitons.