{"title":"Automation et travail","authors":"Bo Harvey","doi":"10.4000/variations.2170","DOIUrl":"https://doi.org/10.4000/variations.2170","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"19 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79683246","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":"Peter Weiss versus Martin Heidegger","authors":"Lucia Sagradini","doi":"10.4000/variations.2238","DOIUrl":"https://doi.org/10.4000/variations.2238","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"96 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75991967","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":"Heureux comme Heidegger en France ?","authors":"Alexander Neumann","doi":"10.4000/variations.2178","DOIUrl":"https://doi.org/10.4000/variations.2178","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"38 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84291568","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":"La paranoïa androïde","authors":"Amelia Horgan","doi":"10.4000/variations.2164","DOIUrl":"https://doi.org/10.4000/variations.2164","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"14 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87607749","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":"En attendant les robots","authors":"Jason Read","doi":"10.4000/variations.2159","DOIUrl":"https://doi.org/10.4000/variations.2159","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"81 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85592239","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":"Pour une critique des approches « média-techniques »","authors":"Fabien Granjon","doi":"10.4000/variations.2240","DOIUrl":"https://doi.org/10.4000/variations.2240","url":null,"abstract":"","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"31 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77002214","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}
Abstract We prove the absence of the Lavrentiev gap for non-autonomous functionals ℱ ( u ) ≔ ∫ Ω f ( x , D u ( x ) ) 𝑑 x , mathcal{F}(u)coloneqqint_{Omega}f(x,Du(x)),dx, where the density f ( x , z ) {f(x,z)} is α-Hölder continuous with respect to x ∈ Ω ⊂ ℝ n {xinOmegasubsetmathbb{R}^{n}} , it satisfies the ( p , q ) {(p,q)} -growth conditions | z | p ⩽ f ( x , z ) ⩽ L ( 1 + | z | q ) , lvert zrvert^{p}leqslant f(x,z)leqslant L(1+lvert zrvert^{q}), where 1 < p < q < p ( n + α n ) {1
摘要证明了非自治函子＿ ＿ (u)是∫Ω f ＿ (x)，D ＿ (u) ＿ (x)) ＿𝑑x, mathcal{F} (u) coloneqqint ＿ {Omega} f(x,Du(x))，dx，其中f ＿ (x, z) {f(x,z)}对于x∈Ω是α-Hölder连续的，∧∈Ω {xinOmegasubsetmathbb{R} ^ {n}}满足(p，q) {(p,q)} -生长条件| z | p≤f(x,z)≤L(1+ | z | q)， lvert z rvert ^ {p}leqslant f(x,z) leqslant L(1+ lvert z rvert ^ {q})，其中1 < p < q < p≠(n + α n) {1
{"title":"No Lavrentiev gap for some double phase integrals","authors":"Filomena De Filippis, F. Leonetti","doi":"10.1515/acv-2021-0109","DOIUrl":"https://doi.org/10.1515/acv-2021-0109","url":null,"abstract":"Abstract We prove the absence of the Lavrentiev gap for non-autonomous functionals ℱ ( u ) ≔ ∫ Ω f ( x , D u ( x ) ) 𝑑 x , mathcal{F}(u)coloneqqint_{Omega}f(x,Du(x)),dx, where the density f ( x , z ) {f(x,z)} is α-Hölder continuous with respect to x ∈ Ω ⊂ ℝ n {xinOmegasubsetmathbb{R}^{n}} , it satisfies the ( p , q ) {(p,q)} -growth conditions | z | p ⩽ f ( x , z ) ⩽ L ( 1 + | z | q ) , lvert zrvert^{p}leqslant f(x,z)leqslant L(1+lvert zrvert^{q}), where 1 < p < q < p ( n + α n ) {1<p<q<p(frac{n+alpha}{n})} , and it can be approximated from below by suitable densities f k {f_{k}} .","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44281893","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}
Abstract We propose two new direct methods for proving partial regularity of solutions of nonlinear elliptic or parabolic systems. The methods are based on two similar interpolation inequalities for solutions of linear systems with constant coefficient. The first results from an interpolation inequality of L p {L^{p}} norms in combination with an L p {L^{p}} estimate with low exponent p > 1 {p>1} . For the second, we provide a functional-analytic proof, that also sheds light upon the A-harmonic approximation lemma of Duzaar and Steffen. Both methods use a Caccioppoli inequality and avoid higher integrability. We illustrate the methods in detail for the case of a quasilinear elliptic system.
提出了两种新的直接证明非线性椭圆型或抛物型系统解的部分正则性的方法。该方法基于常系数线性系统解的两个类似插值不等式。第一个结果是由L p {L^{p}}范数与L p {L^{p}}低指数p> {p>1}估计相结合的插值不等式得到的。其次，我们提供了一个泛函解析证明，这也揭示了Duzaar和Steffen的a调和近似引理。两种方法都使用了Caccioppoli不等式，避免了较高的可积性。对于拟线性椭圆系统，我们详细地说明了这些方法。
{"title":"Interpolation inequalities for partial regularity","authors":"C. Hamburger","doi":"10.1515/acv-2021-0043","DOIUrl":"https://doi.org/10.1515/acv-2021-0043","url":null,"abstract":"Abstract We propose two new direct methods for proving partial regularity of solutions of nonlinear elliptic or parabolic systems. The methods are based on two similar interpolation inequalities for solutions of linear systems with constant coefficient. The first results from an interpolation inequality of L p {L^{p}} norms in combination with an L p {L^{p}} estimate with low exponent p > 1 {p>1} . For the second, we provide a functional-analytic proof, that also sheds light upon the A-harmonic approximation lemma of Duzaar and Steffen. Both methods use a Caccioppoli inequality and avoid higher integrability. We illustrate the methods in detail for the case of a quasilinear elliptic system.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":"16 1","pages":"651 - 663"},"PeriodicalIF":1.7,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44430146","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}
Abstract In this paper, we study the existence and interior W 2 , p {W^{2,p}} -regularity of the solution, and the regularity of the free boundary ∂ { u > ϕ } {partial{u>phi}} to the obstacle problem of the porous medium equation, u t = Δ u m {u_{t}=Delta u^{m}} ( m > 1 {m>1} ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ { u > ϕ } {partial{u>phi}} and ∂ { u > 0 } {partial{u>0}} , we consider two cases on the initial data which make the free boundary ∂ { u > ϕ } {partial{u>phi}} separate from the free boundary ∂ { u > 0 } {partial{u>0}} . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 {C^{1}} -regularity of the free boundary ∂ { u > ϕ } {partial{u>phi}} is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator.
{"title":"Properties of the free boundaries for the obstacle problem of the porous medium equations","authors":"Sunghoon Kim, Ki-ahm Lee, Jinwan Park","doi":"10.1515/acv-2021-0113","DOIUrl":"https://doi.org/10.1515/acv-2021-0113","url":null,"abstract":"Abstract In this paper, we study the existence and interior W 2 , p {W^{2,p}} -regularity of the solution, and the regularity of the free boundary ∂ { u > ϕ } {partial{u>phi}} to the obstacle problem of the porous medium equation, u t = Δ u m {u_{t}=Delta u^{m}} ( m > 1 {m>1} ) with the obstacle function ϕ. The penalization method is applied to have the existence and interior regularity. To deal with the interaction between two free boundaries ∂ { u > ϕ } {partial{u>phi}} and ∂ { u > 0 } {partial{u>0}} , we consider two cases on the initial data which make the free boundary ∂ { u > ϕ } {partial{u>phi}} separate from the free boundary ∂ { u > 0 } {partial{u>0}} . Then the problem is converted into the obstacle problem for a fully nonlinear operator. Hence, the C 1 {C^{1}} -regularity of the free boundary ∂ { u > ϕ } {partial{u>phi}} is obtained by the regularity theory of a class of obstacle problems for the general fully nonlinear operator.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47653952","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}
Abstract In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q 0 ⊂ ℝ d → ℝ {u:Q_{0}subsetmathbb{R}^{d}tomathbb{R}} which are integrable on β-dimensional subsets of the cube Q 0 {Q_{0}} : ∥ u ∥ BMO β ( Q 0 ) := sup Q ⊂ Q 0 inf c ∈ ℝ 1 l ( Q ) β ∫ Q | u - c | 𝑑 ℋ ∞ β , displaystyle|u|_{mathrm{BMO}^{beta}(Q_{0})}vcentcolon=sup_{Qsubset Q_{% 0}}inf_{cinmathbb{R}}frac{1}{l(Q)^{beta}}int_{Q}|u-c|,dmathcal{H}^{% beta}_{infty}, where the supremum is taken over all finite subcubes Q parallel to Q 0 {Q_{0}} , l ( Q ) {l(Q)} is the length of the side of the cube Q, and ℋ ∞ β {mathcal{H}^{beta}_{infty}} is the Hausdorff content. In the case β = d {beta=d} we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every β ∈ ( 0 , d ] {betain(0,d]} one has a dimensionally appropriate analogue of the John–Nirenberg inequality for functions with bounded β-dimensional mean oscillation: There exist constants c , C > 0 {c,C>0} such that ℋ ∞ β ( { x ∈ Q : | u ( x ) - c Q | > t } ) ≤ C l ( Q ) β exp ( - c t ∥ u ∥ BMO β ( Q 0 ) ) displaystylemathcal{H}^{beta}_{infty}({xin Q:|u(x)-c_{Q}|>t})leq Cl(Q)% ^{beta}expbiggl{(}-frac{ct}{|u|_{mathrm{BMO}^{beta}(Q_{0})}}biggr{)} for every t > 0 {t>0} , u ∈ BMO β ( Q 0 ) {uinmathrm{BMO}^{beta}(Q_{0})} , Q ⊂ Q 0 {Qsubset Q_{0}} , and suitable c Q ∈ ℝ {c_{Q}inmathbb{R}} . Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.
{"title":"On functions of bounded β-dimensional mean oscillation","authors":"You-Wei Chen, Daniel Spector","doi":"10.1515/acv-2022-0084","DOIUrl":"https://doi.org/10.1515/acv-2022-0084","url":null,"abstract":"Abstract In this paper, we define a notion of β-dimensional mean oscillation of functions u : Q 0 ⊂ ℝ d → ℝ {u:Q_{0}subsetmathbb{R}^{d}tomathbb{R}} which are integrable on β-dimensional subsets of the cube Q 0 {Q_{0}} : ∥ u ∥ BMO β ( Q 0 ) := sup Q ⊂ Q 0 inf c ∈ ℝ 1 l ( Q ) β ∫ Q | u - c | 𝑑 ℋ ∞ β , displaystyle|u|_{mathrm{BMO}^{beta}(Q_{0})}vcentcolon=sup_{Qsubset Q_{% 0}}inf_{cinmathbb{R}}frac{1}{l(Q)^{beta}}int_{Q}|u-c|,dmathcal{H}^{% beta}_{infty}, where the supremum is taken over all finite subcubes Q parallel to Q 0 {Q_{0}} , l ( Q ) {l(Q)} is the length of the side of the cube Q, and ℋ ∞ β {mathcal{H}^{beta}_{infty}} is the Hausdorff content. In the case β = d {beta=d} we show this definition is equivalent to the classical notion of John and Nirenberg, while our main result is that for every β ∈ ( 0 , d ] {betain(0,d]} one has a dimensionally appropriate analogue of the John–Nirenberg inequality for functions with bounded β-dimensional mean oscillation: There exist constants c , C > 0 {c,C>0} such that ℋ ∞ β ( { x ∈ Q : | u ( x ) - c Q | > t } ) ≤ C l ( Q ) β exp ( - c t ∥ u ∥ BMO β ( Q 0 ) ) displaystylemathcal{H}^{beta}_{infty}({xin Q:|u(x)-c_{Q}|>t})leq Cl(Q)% ^{beta}expbiggl{(}-frac{ct}{|u|_{mathrm{BMO}^{beta}(Q_{0})}}biggr{)} for every t > 0 {t>0} , u ∈ BMO β ( Q 0 ) {uinmathrm{BMO}^{beta}(Q_{0})} , Q ⊂ Q 0 {Qsubset Q_{0}} , and suitable c Q ∈ ℝ {c_{Q}inmathbb{R}} . Our proof relies on the establishment of capacitary analogues of standard results in integration theory that may be of independent interest.","PeriodicalId":49276,"journal":{"name":"Advances in Calculus of Variations","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2022-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44010899","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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