不同单项权等环集的$\varepsilon$ - $\varepsilon$性质及有界性

IF 1.3 3区数学 Q4 AUTOMATION & CONTROL SYSTEMS

Esaim-Control Optimisation and Calculus of Variations Pub Date : 2023-03-31 DOI:10.1051/cocv/2023023

Emerson Abreu, Leandro G. Fernandes, Joel Cruz Ramirez

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引用次数: 0

摘要

考虑一元权$x^{a}=\vert x_{1}\vert^{a_{1}}\ldots\vert x_{N}\vert^{a_{N} $中的每一个$i $都是一个非负实数，在$\mathbb{R} $中，我们建立了$\varepsilon-\varepsilon$的性质和周长和体积具有不同一元权的等围集的有界性。此外，我们还给出了当不可能关联相应的Sobolev不等式时等周不等式不存在的情况。最后，对于N=2，我们发展了一种原始的对称类型，我们称之为星形斯坦纳对称，并将其应用于一类具有不同单项权的等周问题。

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The $\varepsilon$ - $\varepsilon$ property and the boundedness of isoperimetric sets with different monomial weights

We consider a class of monomial weights $x^{A}=\vert x_{1}\vert^{a_{1}}\ldots\vert x_{N}\vert^{a_{N}}$ in $\mathbb{R}^{N}$, where $a_{i}$ is a nonnegative real number for each $i\in\{1,\ldots,N\}$, and we establish the $\varepsilon-\varepsilon$ property and the boundedness of isoperimetric sets with different monomial weights for the perimeter and volume. Moreover, we present cases of nonexistence of the isoperimetric inequality when it is not possible to associate the corresponding Sobolev inequality. Finally, for $N=2$, we developed an original type of symmetrization, which we call star-shaped Steiner symmetrization, and we apply it to a class of isoperimetric problems with different monomial weights.

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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.