自由边界蒙日-安培方程

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS Esaim-Control Optimisation and Calculus of Variations Pub Date : 2022-07-06 DOI:10.1051/cocv/2022048

M. Sedjro

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引用次数: 0

摘要

本文考虑了$\mathbb{R}^2$自由边界域上的一类蒙日-安培方程，其中未知函数的值是在自由边界上规定的。从变分的角度来看，这些方程描述了从一个先验的不确定的源域到一个规定的目标域的最优传输问题。在源域上密度函数的奇异条件下，证明了这些蒙日-安培方程的变分解的存在唯一性。并在一定条件下给出了正则性结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Free boundary Monge-Ampere equations

In this paper, we consider a class of Monge -Ampere equations in a free boundary domain of $\mathbb{R}^2$ where the value of the unknown function is prescribed on the free boundary. From a variational point of view, these equations describe an optimal transport problem from an a priori undetermined source domain to a prescribed target domain. We prove the existence and uniqueness of a variational solution to these Monge -Ampere equations under a singularity condition on the density function on the source domain. Furthermore, we provide regularity results under some conditions on the prescribed domain.

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Esaim-Control Optimisation and Calculus of Variations Mathematics-Computational Mathematics

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7.10%

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期刊介绍： ESAIM: COCV strives to publish rapidly and efficiently papers and surveys in the areas of Control, Optimisation and Calculus of Variations. Articles may be theoretical, computational, or both, and they will cover contemporary subjects with impact in forefront technology, biosciences, materials science, computer vision, continuum physics, decision sciences and other allied disciplines. Targeted topics include: in control: modeling, controllability, optimal control, stabilization, control design, hybrid control, robustness analysis, numerical and computational methods for control, stochastic or deterministic, continuous or discrete control systems, finite-dimensional or infinite-dimensional control systems, geometric control, quantum control, game theory; in optimisation: mathematical programming, large scale systems, stochastic optimisation, combinatorial optimisation, shape optimisation, convex or nonsmooth optimisation, inverse problems, interior point methods, duality methods, numerical methods, convergence and complexity, global optimisation, optimisation and dynamical systems, optimal transport, machine learning, image or signal analysis; in calculus of variations: variational methods for differential equations and Hamiltonian systems, variational inequalities; semicontinuity and convergence, existence and regularity of minimizers and critical points of functionals, relaxation; geometric problems and the use and development of geometric measure theory tools; problems involving randomness; viscosity solutions; numerical methods; homogenization, multiscale and singular perturbation problems.