Advances in Calculus of Variations最新文献

{"title":"Γ-convergence analysis of the nonlinear self-energy induced by edge dislocations in semi-discrete and discrete models in two dimensions","authors":"Roberto Alicandro, Lucia De Luca, Mariapia Palombaro, Marcello Ponsiglione","doi":"10.1515/acv-2023-0053","DOIUrl":"https://doi.org/10.1515/acv-2023-0053","url":null,"abstract":"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 <jats:italic>dilute regime</jats:italic>, we analyze the asymptotic behavior of the nonlinear elastic energy, as the <jats:italic>core-radius</jats:italic> (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.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139927428","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}

{"title":"Another proof of the existence of homothetic solitons of the inverse mean curvature flow","authors":"Shu-Yu Hsu","doi":"10.1515/acv-2022-0092","DOIUrl":"https://doi.org/10.1515/acv-2022-0092","url":null,"abstract":"We will give a new proof of the existence of non-compact homothetic solitons of the inverse mean curvature flow in <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:mi>n</m:mi> </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-2022-0092_eq_0192.png\" /> <jats:tex-math>{mathbb{R}^{n}timesmathbb{R}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0220.png\" /> <jats:tex-math>{ngeq 2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>y</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>r</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0150.png\" /> <jats:tex-math>{(r,y(r))}</jats:tex-math> </jats:alternatives> </jats:inline-formula> or <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>r</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0149.png\" /> <jats:tex-math>{(r(y),y)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>r</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy=\"false\">|</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">|</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0232.png\" /> <jats:tex-math>{r=|x|}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_acv-2022-0092_eq_0257.png\" /> <jats:tex-math>{xinmathbb{R}^{n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, is the radially symmetric coordinate and <jats:inline-formula>","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139586295","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}

{"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":null,"pages":null},"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}

{"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>&lt;</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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"title":"Monotonicity of entire solutions to reaction-diffusion equations involving fractional p-Laplacian","authors":"Qing Guo","doi":"10.1515/acv-2022-0109","DOIUrl":"https://doi.org/10.1515/acv-2022-0109","url":null,"abstract":"We obtain the one-dimensional symmetry and monotonicity of the entire positive solutions to some reaction-diffusion equations involving fractional <jats:italic>p</jats:italic>-Laplacian by virtue of the sliding method. More precisely, we consider the following problem <jats:disp-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>{</m:mo> <m:mtable columnspacing=\"0pt\" displaystyle=\"true\" rowspacing=\"0pt\"> <m:mtr> <m:mtd columnalign=\"right\"> <m:mrow> <m:mrow> <m:mfrac> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo>∂</m:mo> <m:mo>⁡</m:mo> <m:mi>t</m:mi> </m:mrow> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo>+</m:mo> <m:mrow> <m:msubsup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mo>-</m:mo> <m:mi mathvariant=\"normal\">Δ</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mi>p</m:mi> <m:mi>s</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign=\"left\"> <m:mrow> <m:mrow> <m:mi /> <m:mo>=</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign=\"left\"> <m:mrow> <m:mo lspace=\"12.5pt\" stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign=\"right\"> <m:mrow> <m:mrow> <m:mi /> <m:mo>∈</m:mo> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>×</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"right\"> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:mtd> <m:mtd columnalign=\"left\"> <m:mrow> <m:mrow> <m:mi /> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign=\"left\"> <m:mrow> <m:mo lspace=\"12.5pt\" stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign=\"right\"> <m:mrow> <m:mrow> <m:mi /> <m:mo>∈</m:mo> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>×</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign=\"right\"> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>t</m:mi> <m:mo stretch","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139421623","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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}