Advances in Calculus of Variations最新文献_第2页

Γ-convergence analysis of the nonlinear self-energy induced by edge dislocations in semi-discrete and discrete models in two dimensions 二维半离散和离散模型中边缘位错诱发的非线性自能的Γ-收敛分析

IF 1.7 3区数学 Q1 Mathematics

Advances in Calculus of Variations

Pub Date : 2024-02-20 DOI: 10.1515/acv-2023-0053

Roberto Alicandro, Lucia De Luca, Mariapia Palombaro, Marcello Ponsiglione

We propose nonlinear semi-discrete and discrete models for the elastic energy induced by a finite system of edge dislocations in two dimensions. Within the dilute regime, we analyze the asymptotic behavior of the nonlinear elastic energy, as the core-radius (in the semi-discrete model) and the lattice spacing (in the purely discrete one) vanish. Our analysis passes through a linearization procedure within the rigorous framework of Γ-convergence.

我们提出了二维有限边缘位错系统诱导弹性能量的非线性半离散和离散模型。在稀释体系中，我们分析了非线性弹性能量的渐近行为，即核心半径（在半离散模型中）和晶格间距（在纯离散模型中）消失。我们的分析是在Γ收敛的严格框架内通过线性化程序进行的。

引用次数: 0

Another proof of the existence of homothetic solitons of the inverse mean curvature flow 反均值曲率流同向孤子存在的另一个证明

IF 1.7 3区数学 Q1 Mathematics

Advances in Calculus of Variations

Pub Date : 2024-01-30 DOI: 10.1515/acv-2022-0092

Shu-Yu Hsu

We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow in ℝ n × ℝ

{mathbb{R}^{n}timesmathbb{R}}

, n ≥ 2

{ngeq 2}

, of the form ( r , y ⁢ ( r ) )

{(r,y(r))}

or ( r ⁢ ( y ) , y )

{(r(y),y)}

, where r = | x |

{r=|x|}

, x ∈ ℝ n

{xinmathbb{R}^{n}}

, is the radially symmetric coordinate and

我们将给出一个新的证明：在 ℝ n × ℝ {mathbb{R}^{n}timesmathbb{R}} 中存在反均值曲率流的非紧凑同调孤子。 , n ≥ 2 {ngeq 2} , 形式为 ( r , y ( r ) ) {(r,y(r))} 或 ( r ( y ) , y ) {(r(y),y)} , 其中 r = | x | {r=|x|} x ∈ ℝ n {xinmathbb{R}^{n}} ，是径向对称坐标。是径向对称坐标，y∈ ℝ {yinmathbb{R}} 。 .更确切地说，对于任意 1 n < λ < 1 n - 1 {frac{1}{n}<lambda<frac{1}{n-1}} 和 μ < 0 {mu<0} ，我们将给出新的证明。我们将给出一个新的证明，证明存在一个唯一的解 r ( y ) ∈ C 2 ( μ , ∞ ) ∩ C ( [ μ , ∞ ) ) {r(y)in C^{2}(mu,infty)cap C([mu,infty))}的方程 r y y ( y ) 1 + r y ( y ) 2 = n - 1 r ( y ) - 1 + r y ( y ) 2 λ ( r ( y ) - y r y ( y ) ) , r ( y ) > 0 , frac{r_{yy}(y)}{1+r_{y}(y)^{2}}=frac{n-1}{r(y)}-frac{1+r_{y}(y)^{2}}{% lambda(r(y)-yr_{y}(y))},quad r(y)>；0, in ( μ , ∞ ) {(mu,infty)} which satisfies r ( μ ) = 0 {r(mu)=0} and r y ( μ ) = lim y ↘ μ r y ( y ) = + ∞ {r_{y}(mu)=lim_{ysearrowmu}r_{y}(y)=+infty} .我们证明存在常数 y 2 > y 1 > 0 y_{2}>y_{1}>0，使得 r y ( y ) > 0 r_{y}(y)>0 for any μ < y < y 1 mu<y<y_{1}。 , r y ( y 1 ) = 0 r_{y}(y_{1})=0 , r y ( y ) < 0 r_{y}(y)<0 for any y > y 1 y>y_{1} , r y y ( y ) < 0 r_{y}(y)<0 for any y > y 1 y>y_{1} , r y y ( y ) < 0 r_{yy}(y)<0 for any μ < y < y 2 mu<y<y_{2} , r y y ( y ) < 0 r_{yy}(y)<0 for any μ < y < y 2 对于任意 y > y 2 {y>y_{2}} ，r y y ( y 2 ) = 0 r_{yy}(y_{2})=0 且 r y y ( y ) > 0 {r_{yy}(y)>0} 。 .此外，lim y → + ∞ r ( y ) = 0 {lim_{yto+infty}r(y)=0} 和 lim y → + ∞ y r y ( y ) = 0 {lim_{yto+infty}yr_{y}(y)=0} .

引用次数: 0

Non-local BV functions and a denoising model with L 1 fidelity 非局部 BV 函数和保真度为 L 1 的去噪模型

IF 1.7 3区数学 Q1 Mathematics

Advances in Calculus of Variations

Pub Date : 2024-01-30 DOI: 10.1515/acv-2023-0082

Konstantinos Bessas, Giorgio Stefani

We study a general total variation denoising model with weighted L 1

{L^{1}}

fidelity, where the regularizing term is a non-local variation induced by a suitable (non-integrable) kernel K, and the approximation term is given by the L 1

{L^{1}}

norm with respect to a non-singular measure with positively lower-bounded L ∞

{L^{infty}}

density. We provide a detailed analysis of the space of non-local BV

mathrm{BV}

functions with finite total K-variation, with special emphasis on compactness, Lusin-type estimates, Sobolev embeddings and isoperimetric and monotonicity properties of the K-variation and the associated K-perimeter. Finally, we deal with the theory of Cheeger sets in this non-local setting and we apply it to the study of the fidelity in our model.

我们研究了一种具有加权 L 1 {L^{1}}保真度的一般总变异去噪模型，其中正则化项是由合适的（不可积分的）核 K 引起的非局部变异，而逼近项则由相对于具有正下限 L ∞ {L^{infty}} 密度的非邢性度量的 L 1 {L^{1}}规范给出。我们详细分析了具有有限总 K 变量的非局部 BV mathrm{BV} 函数空间，特别强调了 K 变量和相关 K 周长的紧凑性、Lusin 型估计、Sobolev 嵌入和等周性与单调性。最后，我们将讨论这种非局部设置中的切格集理论，并将其应用于我们模型中保真度的研究。

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引用次数: 0

Quasiconformal, Lipschitz, and BV mappings in metric spaces 度量空间中的准共形、Lipschitz 和 BV 映射

IF 1.7 3区数学 Q1 Mathematics

Advances in Calculus of Variations

Pub Date : 2024-01-12 DOI: 10.1515/acv-2022-0071

Panu Lahti

Consider a mapping f : X → Y

{fcolon Xto Y}

between two metric measure spaces. We study generalized versions of the local Lipschitz number Lip ⁡ f

{operatorname{Lip}f}

, as well as of the distortion number H f

{H_{f}}

that is used to define quasiconformal mappings. Using these numbers, we give sufficient conditions for f being a BV mapping f ∈ BV loc ⁢ ( X ; Y )

{finmathrm{BV}_{mathrm{loc}}(X;Y)}

or a Newton–Sobolev mapping f ∈ N loc 1 , p ⁢ ( X ; Y )

{fin N_{mathrm{loc}}^{1,p}(X;Y)}

, with 1 ≤ p < ∞

考虑两个度量空间之间的映射 f : X → Y {fcolon Xto Y} 。我们研究局部 Lipschitz 数 Lip f {operatorname{Lip}f} 的广义版本，以及用于定义准共形映射的变形数 H f {H_{f}} 的广义版本。利用这些数字，我们给出了 f 是 BV 映射 f∈ BV loc ( X ; Y ) {finmathrm{BV}_{mathrm{loc}}(X. Y)} 或牛顿映射 f∈ BV loc ( X ; Y ) {finmathrm{BV}_{mathrm{loc}}(X. Y)} 的充分条件；Y)} 或者牛顿-索博列夫映射 f∈ N loc 1 , p ( X ; Y ) {fin N_{mathrm{loc}}^{1,p}(X;Y)} , 其中 1 ≤ p < ∞ {1leq p<infty} 。

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引用次数: 0

Generalized minimizing movements for the varifold Canham–Helfrich flow 变分卡纳姆-赫尔弗里希流的广义最小化运动

IF 1.7 3区数学 Q1 Mathematics

Advances in Calculus of Variations

Pub Date : 2024-01-12 DOI: 10.1515/acv-2022-0056

Katharina Brazda, Martin Kružík, Ulisse Stefanelli

The gradient flow of the Canham–Helfrich functional is tackled via the generalized minimizing movements approach. We prove the existence of solutions in Wasserstein spaces of varifolds, as well as upper and lower diameter bounds. In the more regular setting of multiply covered C 1 , 1

{C^{1,1}}

surfaces, we provide a Li–Yau-type estimate for the Canham–Helfrich energy and prove the conservation of multiplicity along the evolution.

Canham-Helfrich 函数的梯度流是通过广义最小化运动方法解决的。我们证明了变曲率的瓦瑟斯坦空间中解的存在性，以及直径的上下限。在多面覆盖的 C 1 , 1 {C^{1,1}} 曲面这一更为常规的环境中，我们提供了 Canham-Helfrich 能量的 Li-Yau 型估计，并证明了沿演化过程的多重性守恒。

引用次数: 0

Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure: The limiting regimes 具有周期性微观结构的完全塑性板材的有效准静态演化模型：极限状态

IF 1.7 3区数学 Q1 Mathematics

Advances in Calculus of Variations

Pub Date : 2024-01-11 DOI: 10.1515/acv-2023-0020

Marin Bužančić, Elisa Davoli, Igor Velčić

We identify effective models for thin, linearly elastic and perfectly plastic plates exhibiting a microstructure resulting from the periodic alternation of two elastoplastic phases. We study here both the case in which the thickness of the plate converges to zero on a much faster scale than the periodicity parameter and the opposite scenario in which homogenization occurs on a much finer scale than dimension reduction. After performing a static analysis of the problem, we show convergence of the corresponding quasistatic evolutions. The methodology relies on two-scale convergence and periodic unfolding, combined with suitable measure-disintegration results and evolutionary Γ-convergence.

我们确定了线性弹性和完全塑性薄板的有效模型，这些薄板的微观结构是由两个弹塑性相周期性交替产生的。在这里，我们既研究了板厚度以比周期参数更快的速度趋近于零的情况，也研究了与之相反的情况，即均质化以比尺寸缩小更小的尺度发生。在对问题进行静态分析后，我们展示了相应准静态演化的收敛性。该方法依赖于双尺度收敛和周期性展开，并结合了合适的度量分解结果和演化Γ收敛。

引用次数: 0

Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian 涉及分数 p-Laplacian 的反应扩散方程全解的单调性

IF 1.7 3区数学 Q1 Mathematics

Advances in Calculus of Variations

Pub Date : 2024-01-10 DOI: 10.1515/acv-2022-0109

Qing Guo

We obtain the one-dimensional symmetry and monotonicity of the entire positive solutions to some reaction-diffusion equations involving fractional p-Laplacian by virtue of the sliding method. More precisely, we consider the following problem { ∂ ⁡ u ∂ ⁡ t ⁢ ( x , t ) + ( - Δ ) p s ⁢ u ⁢ ( x , t ) = f ⁢ ( t , u ⁢ ( x , t ) ) , ( x , t ) ∈ Ω × ℝ , u ⁢ ( x , t ) > 0 , ( x , t ) ∈ Ω × ℝ , u ⁢ ( x , t

我们利用滑动法得到了一些涉及分数 p-Laplacian 的反应扩散方程全正解的一维对称性和单调性。更确切地说，我们考虑以下问题 { ∂ u ∂ t ( x , t ) + ( - Δ ) p s u ( x , t ) = f ( t , u ( x , t ) ) , ( x , t ) ∈ Ω × ℝ , u ( x , t ) > 0 , ( x , t ) ∈ Ω × ℝ , u ( x , t ) = 0 , ( x , t ) ∈ Ω c × ℝ , left{begin{aligned}displaystyle{}frac{partial u}{partial t}(x,t)+(-% Delta)_{p}^{s}u(x,t)&displaystyle=f(t,u(x,t)),&hskip 10.0pt(x,t)&%displaystyleinOmegatimesmathbb{R},displaystyle u(x,t)&displaystyle>0,&hskip 10.0pt(x,t)&displaystylein% Omegatimesmathbb{R},displaystyle u(x,t)&displaystyle=0,&hskip 10.0pt(x,t)&displaystylein% Omega^{c}timesmathbb{R},end{aligned}right. 其中 s∈ ( 0 , 1 ) {sin(0,1)} , p≥ 2 {pgeq 2} , ( - Δ ) p s {(-Delta)_{p}^{s}} 是分数 p-拉普拉奇函数，f ( t , u ) {f(t、u)} 是某个连续函数，域 Ω ⊂ n {Omegasubsetmathbb{R}^{n} 是无界的，且 Ω c = ℝ n ∖ Ω {Omega^{c}=mathbb{R}^{n}setminusOmega} 。首先，我们建立一个涉及抛物线 p-Laplacian 算子的最大值原理。然后，在 f 的特定条件下，我们证明了 t∈ ℝ {tinmathbb{R}} 中均匀远离边界的解的渐近行为。 .最后，利用滑动方法推导出有界正全解的单调性和唯一性。据我们所知，之前还没有任何关于抛物分式 p-Laplacian 方程解的对称性和单调性的结果。

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引用次数: 0

Asymptotic analysis of single-slip crystal plasticity in the limit of vanishing thickness and rigid elasticity 厚度和刚性弹性消失极限下单滑晶塑性的渐近分析

IF 1.7 3区数学 Q1 Mathematics

Advances in Calculus of Variations

Pub Date : 2024-01-10 DOI: 10.1515/acv-2023-0009

Dominik Engl, Stefan Krömer, Martin Kružík

We perform via Γ-convergence a 2d-1d dimension reduction analysis of a single-slip elastoplastic body in large deformations. Rigid plastic and elastoplastic regimes are considered. In particular, we show that limit deformations can essentially freely bend even if subjected to the most restrictive constraints corresponding to the elastically rigid single-slip regime. The primary challenge arises in the upper bound where the differential constraints render any bending without incurring an additional energy cost particularly difficult. We overcome this obstacle with suitable non-smooth constructions and prove that a Lavrentiev phenomenon occurs if we artificially restrict our model to smooth deformations. This issue is absent if the differential constraints are appropriately softened.

我们通过Γ-收敛对大变形中的单滑动弹塑性体进行 2d-1d 维度缩减分析。我们考虑了刚性塑性和弹塑性状态。我们特别指出，即使受到与弹性刚性单滑移状态相对应的最严格约束，极限变形基本上也能自由弯曲。主要的挑战出现在上界，在这里，差分约束使得在不产生额外能量成本的情况下进行任何弯曲都特别困难。我们利用合适的非光滑结构克服了这一障碍，并证明如果我们人为地将模型限制为光滑变形，就会出现拉夫伦捷夫现象。如果适当软化微分约束，则不会出现这一问题。

引用次数: 0

Least energy solutions for affine p-Laplace equations involving subcritical and critical nonlinearities 涉及亚临界和临界非线性的仿射 p-Laplace 方程的最小能量解

IF 1.7 3区数学 Q1 Mathematics

Advances in Calculus of Variations

Pub Date : 2024-01-10 DOI: 10.1515/acv-2022-0050

Edir Júnior Ferreira Leite, Marcos Montenegro

The paper is concerned with Lane–Emden and Brezis–Nirenberg problems involving the affine p-Laplace nonlocal operator Δ p 𝒜

{Delta_{p}^{cal A}}

, which has been introduced in [J. Haddad, C. H. Jiménez and M. Montenegro, From affine Poincaré inequalities to affine spectral inequalities, Adv. Math. 386 2021, Article ID 107808] driven by the affine L p

{L^{p}}

energy ℰ p , Ω

{{cal E}_{p,Omega}}

from convex geometry due to [E. Lutwak, D. Yang and G. Zhang, Sharp affine L p

L_{p}

Sobolev inequalities, J. Differential Geom. 62 2002, 1, 17–38]. We are particularly interested in the existence and nonexistence of positive C 1

{C^{1}}

solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of ℰ p , Ω

{

本文涉及涉及仿射 p-Laplace 非局部算子 Δ p 𝒜 {Delta_{p}^{cal A}} 的 Lane-Emden 和 Brezis-Nirenberg 问题。 , which has been introduced in [J. Haddad, C. H. Jiménez and M. Montenegro, From affine Poincaré inequalities to affine spectral inequalities, Adv. Math.386 2021, Article ID 107808] 由凸几何中的仿射 L p {L^{p}} 能量 ℰ p , Ω {{cal E}_{p,Omega}} 驱动，归因于 [E. Lutwak, D. Yang.Lutwak, D. Yang and G. Zhang, Sharp affine L p L_{p}. Sobolev 不等式, J. Differential Geom.62 2002, 1, 17-38].我们对最小能量型正 C 1 {C^{1}} 解的存在与不存在特别感兴趣。部分主要困难是由ℰ p , Ω {{cal E}_{p,Omega}} 的不凸性和比较 ℰ p , Ω ( u ) ≤ ∥ u ∥ W 0 1 , p ( Ω ) {{cal E}_{p,Omega}(u)leq|u|_{W^{1,p}_{0}(Omega)}} 一般严格造成的。

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引用次数: 0

The 2+1-convex hull of a~finite set 无穷集的 2+1 凸体

IF 1.7 3区数学 Q1 Mathematics

Advances in Calculus of Variations

Pub Date : 2024-01-01 DOI: 10.1515/acv-2023-0077

Pablo Angulo, Carlos García-Gutiérrez

Rank-one convexity is a weak form of convexity related to convex integration and the elusive notion of quasiconvexity, but more amenable both in theory and practice. However, exact algorithms for computing the rank one convex hull of a finite set are only known for some special cases of separate convexity with a finite number of directions. Both inner approximations either with laminates or T 4

{T_{4}}

’s and outer approximations through polyconvexity are known to be insufficient in general. We study ℝ 2 ⊕ ℝ

{mathbb{R}^{2}oplusmathbb{R}}

-separately convex hulls of finite sets, which is a special case of rank-one convexity with infinitely many directions in which T 4

{T_{4}}

’s are known not to capture the rank one convex hull. When ℝ 3

{mathbb{R}^{3}}

is identified with a subset of 2 × 3

{2times 3}

matrices, it is known to correspond also to quasiconvexity. We propose new inner and outer approximations built upon systematic use of known results, and prove that they agree. The inner approximation allows to understand better the structure of the rank one convex hull. The outer approximation gives rise to a computational algorithm which, in some cases, computes the hull exactly, and in general builds a sequence that c

秩一凸性是凸性的一种弱形式，与凸积分和难以捉摸的准凸性概念有关，但在理论和实践上都更容易理解。然而，计算有限集合的秩一凸壳的精确算法只适用于具有有限方向数的独立凸性的某些特殊情况。无论是使用层状结构或 T 4 {T_{4}} 的内近似，还是通过矩阵的外近似，都需要对有限集合的秩一凸壳进行计算。的内近似和通过多凸性的外近似在一般情况下都是不够的。我们研究了 ⊕ ℝ 2 ⊕ ℝ {mathbb{R}^{2}oplusmathbb{R}} -有限集的分离凸壳，这是具有无限多方向的秩一凸性的一个特例，其中 T 4 {T_{4}} '已知不能捕捉到秩一凸体。当 ℝ 3 {mathbb{R}^{3}} 与 2 × 3 {2times 3} 矩阵的子集确定时，已知它也对应于准凸性。我们在系统利用已知结果的基础上提出了新的内近似和外近似，并证明它们是一致的。通过内近似，可以更好地理解秩一凸壳的结构。外近似产生了一种计算算法，在某些情况下，它能精确计算出凸壳，而在一般情况下，它能建立一个收敛到凸壳的序列。我们使用并系统化了以前计算 D- 凸体的所有尝试，并带来了可能有助于计算一般 D- 凸体的新思路。

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引用次数: 0

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