We study a general total variation denoising model with weighted L1{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 L1{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 BVmathrm{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。
{"title":"Non-local BV functions and a denoising model with L 1 fidelity","authors":"Konstantinos Bessas, Giorgio Stefani","doi":"10.1515/acv-2023-0082","DOIUrl":"https://doi.org/10.1515/acv-2023-0082","url":null,"abstract":"We study a general total variation denoising model with weighted <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0082_eq_0326.png\" /> <jats:tex-math>{L^{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> fidelity, where the regularizing term is a non-local variation induced by a suitable (non-integrable) kernel <jats:italic>K</jats:italic>, and the approximation term is given by the <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0082_eq_0326.png\" /> <jats:tex-math>{L^{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> norm with respect to a non-singular measure with positively lower-bounded <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant=\"normal\">∞</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0082_eq_0328.png\" /> <jats:tex-math>{L^{infty}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> density. We provide a detailed analysis of the space of non-local <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>BV</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0082_eq_0210.png\" /> <jats:tex-math>mathrm{BV}</jats:tex-math> </jats:alternatives> </jats:inline-formula> functions with finite total <jats:italic>K</jats:italic>-variation, with special emphasis on compactness, Lusin-type estimates, Sobolev embeddings and isoperimetric and monotonicity properties of the <jats:italic>K</jats:italic>-variation and the associated <jats:italic>K</jats:italic>-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.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"1 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider a mapping <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>Y</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0071_eq_0735.png" /> <jats:tex-math>{fcolon Xto Y}</jats:tex-math> </jats:alternatives> </jats:inline-formula> between two metric measure spaces. We study generalized versions of the local Lipschitz number <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Lip</m:mi> <m:mo></m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0071_eq_0649.png" /> <jats:tex-math>{operatorname{Lip}f}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, as well as of the distortion number <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>H</m:mi> <m:mi>f</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0071_eq_0480.png" /> <jats:tex-math>{H_{f}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> that is used to define quasiconformal mappings. Using these numbers, we give sufficient conditions for <jats:italic>f</jats:italic> being a BV mapping <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>BV</m:mi> <m:mi>loc</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mi>Y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0071_eq_0759.png" /> <jats:tex-math>{finmathrm{BV}_{mathrm{loc}}(X;Y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or a Newton–Sobolev mapping <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msubsup> <m:mi>N</m:mi> <m:mi>loc</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mi>Y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0071_eq_0751.png" /> <jats:tex-math>{fin N_{mathrm{loc}}^{1,p}(X;Y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic
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} 。
{"title":"Quasiconformal, Lipschitz, and BV mappings in metric spaces","authors":"Panu Lahti","doi":"10.1515/acv-2022-0071","DOIUrl":"https://doi.org/10.1515/acv-2022-0071","url":null,"abstract":"Consider a mapping <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:mi>Y</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0071_eq_0735.png\" /> <jats:tex-math>{fcolon Xto Y}</jats:tex-math> </jats:alternatives> </jats:inline-formula> between two metric measure spaces. We study generalized versions of the local Lipschitz number <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>Lip</m:mi> <m:mo></m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0071_eq_0649.png\" /> <jats:tex-math>{operatorname{Lip}f}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, as well as of the distortion number <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>H</m:mi> <m:mi>f</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0071_eq_0480.png\" /> <jats:tex-math>{H_{f}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> that is used to define quasiconformal mappings. Using these numbers, we give sufficient conditions for <jats:italic>f</jats:italic> being a BV mapping <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msub> <m:mi>BV</m:mi> <m:mi>loc</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mi>Y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0071_eq_0759.png\" /> <jats:tex-math>{finmathrm{BV}_{mathrm{loc}}(X;Y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or a Newton–Sobolev mapping <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msubsup> <m:mi>N</m:mi> <m:mi>loc</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>X</m:mi> <m:mo>;</m:mo> <m:mi>Y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0071_eq_0751.png\" /> <jats:tex-math>{fin N_{mathrm{loc}}^{1,p}(X;Y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:mi>p</m:mi> <m:mo><</m:mo> <m:mi mathvariant=\"normal\">∞</m:mi> </m:mrow> </m:math> <jats:inline-graphic","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"45 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139459397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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 C1,1{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.
{"title":"Generalized minimizing movements for the varifold Canham–Helfrich flow","authors":"Katharina Brazda, Martin Kružík, Ulisse Stefanelli","doi":"10.1515/acv-2022-0056","DOIUrl":"https://doi.org/10.1515/acv-2022-0056","url":null,"abstract":"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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0056_eq_0274.png\" /> <jats:tex-math>{C^{1,1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> surfaces, we provide a Li–Yau-type estimate for the Canham–Helfrich energy and prove the conservation of multiplicity along the evolution.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"25 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139458875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Effective quasistatic evolution models for perfectly plastic plates with periodic microstructure: The limiting regimes","authors":"Marin Bužančić, Elisa Davoli, Igor Velčić","doi":"10.1515/acv-2023-0020","DOIUrl":"https://doi.org/10.1515/acv-2023-0020","url":null,"abstract":"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.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"3 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139458938","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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。
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.
{"title":"Asymptotic analysis of single-slip crystal plasticity in the limit of vanishing thickness and rigid elasticity","authors":"Dominik Engl, Stefan Krömer, Martin Kružík","doi":"10.1515/acv-2023-0009","DOIUrl":"https://doi.org/10.1515/acv-2023-0009","url":null,"abstract":"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.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"85 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139421628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper is concerned with Lane–Emden and Brezis–Nirenberg problems involving the affine <jats:italic>p</jats:italic>-Laplace nonlocal operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="normal">Δ</m:mi> <m:mi>p</m:mi> <m:mi mathvariant="script">𝒜</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0050_eq_0160.png" /> <jats:tex-math>{Delta_{p}^{cal A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, 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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0050_eq_0140.png" /> <jats:tex-math>{L^{p}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> energy <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">ℰ</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0050_eq_0333.png" /> <jats:tex-math>{{cal E}_{p,Omega}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> from convex geometry due to [E. Lutwak, D. Yang and G. Zhang, Sharp affine <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0050_eq_0108.png" /> <jats:tex-math>L_{p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> Sobolev inequalities, J. Differential Geom. 62 2002, 1, 17–38]. We are particularly interested in the existence and nonexistence of positive <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0050_eq_0128.png" /> <jats:tex-math>{C^{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">ℰ</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">Ω</m:mi> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2022-0050_eq_0333.png" /> <jats:tex-math>{
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。
{"title":"Least energy solutions for affine p-Laplace equations involving subcritical and critical nonlinearities","authors":"Edir Júnior Ferreira Leite, Marcos Montenegro","doi":"10.1515/acv-2022-0050","DOIUrl":"https://doi.org/10.1515/acv-2022-0050","url":null,"abstract":"The paper is concerned with Lane–Emden and Brezis–Nirenberg problems involving the affine <jats:italic>p</jats:italic>-Laplace nonlocal operator <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>p</m:mi> <m:mi mathvariant=\"script\">𝒜</m:mi> </m:msubsup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0050_eq_0160.png\" /> <jats:tex-math>{Delta_{p}^{cal A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, 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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0050_eq_0140.png\" /> <jats:tex-math>{L^{p}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> energy <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"script\">ℰ</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0050_eq_0333.png\" /> <jats:tex-math>{{cal E}_{p,Omega}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> from convex geometry due to [E. Lutwak, D. Yang and G. Zhang, Sharp affine <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0050_eq_0108.png\" /> <jats:tex-math>L_{p}</jats:tex-math> </jats:alternatives> </jats:inline-formula> Sobolev inequalities, J. Differential Geom. 62 2002, 1, 17–38]. We are particularly interested in the existence and nonexistence of positive <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0050_eq_0128.png\" /> <jats:tex-math>{C^{1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> solutions of least energy type. Part of the main difficulties are caused by the absence of convexity of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"script\">ℰ</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0050_eq_0333.png\" /> <jats:tex-math>{","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"118 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139421631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>T</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2023-0077_eq_0331.png" /> <jats:tex-math>{T_{4}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>’s and outer approximations through polyconvexity are known to be insufficient in general. We study <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⊕</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2023-0077_eq_0351.png" /> <jats:tex-math>{mathbb{R}^{2}oplusmathbb{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-separately convex hulls of finite sets, which is a special case of rank-one convexity with infinitely many directions in which <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>T</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2023-0077_eq_0331.png" /> <jats:tex-math>{T_{4}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>’s are known not to capture the rank one convex hull. When <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2023-0077_eq_0353.png" /> <jats:tex-math>{mathbb{R}^{3}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is identified with a subset of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>×</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_acv-2023-0077_eq_0130.png" /> <jats:tex-math>{2times 3}</jats:tex-math> </jats:alternatives> </jats:inline-formula> 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
{"title":"The 2+1-convex hull of a~finite set","authors":"Pablo Angulo, Carlos García-Gutiérrez","doi":"10.1515/acv-2023-0077","DOIUrl":"https://doi.org/10.1515/acv-2023-0077","url":null,"abstract":"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 <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0077_eq_0331.png\" /> <jats:tex-math>{T_{4}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>’s and outer approximations through polyconvexity are known to be insufficient in general. We study <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo>⊕</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0077_eq_0351.png\" /> <jats:tex-math>{mathbb{R}^{2}oplusmathbb{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-separately convex hulls of finite sets, which is a special case of rank-one convexity with infinitely many directions in which <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>T</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0077_eq_0331.png\" /> <jats:tex-math>{T_{4}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>’s are known not to capture the rank one convex hull. When <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℝ</m:mi> <m:mn>3</m:mn> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0077_eq_0353.png\" /> <jats:tex-math>{mathbb{R}^{3}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is identified with a subset of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>2</m:mn> <m:mo>×</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2023-0077_eq_0130.png\" /> <jats:tex-math>{2times 3}</jats:tex-math> </jats:alternatives> </jats:inline-formula> 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","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"21 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139077263","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note we prove a sub-Riemannian maximum modulus theorem in a Carnot group. Using a nontrivial counterexample, we also show that such result is best possible, in the sense that in its statement one cannot replace the right-invariant horizontal gradient with the left-invariant one.
{"title":"A sub-Riemannian maximum modulus theorem","authors":"Federico Buseghin, Nicolò Forcillo, Nicola Garofalo","doi":"10.1515/acv-2023-0066","DOIUrl":"https://doi.org/10.1515/acv-2023-0066","url":null,"abstract":"In this note we prove a sub-Riemannian maximum modulus theorem in a Carnot group. Using a nontrivial counterexample, we also show that such result is best possible, in the sense that in its statement one cannot replace the right-invariant horizontal gradient with the left-invariant one.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"55 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139080255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Stefano Biagi, Francesco Esposito, Luigi Montoro, Eugenio Vecchi
We consider positive singular solutions (i.e. with a non-removable singularity) of a system of PDEs driven by p-Laplacian operators and with the additional presence of a nonlinear first order term. By a careful use of a rather new version of the moving plane method, we prove the symmetry of the solutions. The result is already new in the scalar case.
{"title":"Symmetry and monotonicity of singular solutions to p-Laplacian systems involving a first order term","authors":"Stefano Biagi, Francesco Esposito, Luigi Montoro, Eugenio Vecchi","doi":"10.1515/acv-2023-0043","DOIUrl":"https://doi.org/10.1515/acv-2023-0043","url":null,"abstract":"We consider positive singular solutions (i.e. with a non-removable singularity) of a system of PDEs driven by <jats:italic>p</jats:italic>-Laplacian operators and with the additional presence of a nonlinear first order term. By a careful use of a rather new version of the moving plane method, we prove the symmetry of the solutions. The result is already new in the scalar case.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"23 13","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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