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Quantitative Homogenization for the Obstacle Problem and Its Free Boundary. 障碍问题及其自由边界的定量均质化
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-01-01 Epub Date: 2024-08-18 DOI: 10.1007/s00205-024-02015-6
Gohar Aleksanyan, Tuomo Kuusi
{"title":"Quantitative Homogenization for the Obstacle Problem and Its Free Boundary.","authors":"Gohar Aleksanyan, Tuomo Kuusi","doi":"10.1007/s00205-024-02015-6","DOIUrl":"https://doi.org/10.1007/s00205-024-02015-6","url":null,"abstract":"<p><p>In this manuscript we prove quantitative homogenization results for the obstacle problem with bounded measurable coefficients. As a consequence, large-scale regularity results both for the solution and the free boundary for the heterogeneous obstacle problem are derived.</p>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11330955/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142010016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Parabolic Boundary Harnack Inequalities with Right-Hand Side. 带右侧的抛物线边界哈纳克不等式
IF 2.6 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2024-01-01 Epub Date: 2024-08-26 DOI: 10.1007/s00205-024-02017-4
Clara Torres-Latorre
{"title":"Parabolic Boundary Harnack Inequalities with Right-Hand Side.","authors":"Clara Torres-Latorre","doi":"10.1007/s00205-024-02017-4","DOIUrl":"https://doi.org/10.1007/s00205-024-02017-4","url":null,"abstract":"<p><p>We prove the parabolic boundary Harnack inequality in parabolic flat Lipschitz domains by blow-up techniques, allowing, for the first time, a non-zero right-hand side. Our method allows us to treat solutions to equations driven by non-divergence form operators with bounded measurable coefficients, and a right-hand side <math><mrow><mi>f</mi> <mo>∈</mo> <msup><mi>L</mi> <mi>q</mi></msup> </mrow> </math> for <math><mrow><mi>q</mi> <mo>></mo> <mi>n</mi> <mo>+</mo> <mn>2</mn></mrow> </math> . In the case of the heat equation, we also show the optimal <math><msup><mi>C</mi> <mrow><mn>1</mn> <mo>-</mo> <mi>ε</mi></mrow> </msup> </math> regularity of the quotient. As a corollary, we obtain a new way to prove that flat Lipschitz free boundaries are <math><msup><mi>C</mi> <mrow><mn>1</mn> <mo>,</mo> <mi>α</mi></mrow> </msup> </math> in the parabolic obstacle problem and in the parabolic Signorini problem.</p>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC11347492/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142116908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Localized Big Bang Stability for the Einstein-Scalar Field Equations 爱因斯坦-斯卡拉场方程的局部大爆炸稳定性
IF 2.5 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2023-12-08 DOI: 10.1007/s00205-023-01939-9
Florian Beyer, Todd A. Oliynyk
{"title":"Localized Big Bang Stability for the Einstein-Scalar Field Equations","authors":"Florian Beyer,&nbsp;Todd A. Oliynyk","doi":"10.1007/s00205-023-01939-9","DOIUrl":"10.1007/s00205-023-01939-9","url":null,"abstract":"<div><p>We prove the nonlinear stability in the contracting direction of Friedmann–Lemaître–Robertson–Walker (FLRW) solutions to the Einstein-scalar field equations in <span>(nge 3)</span> spacetime dimensions that are defined on spacetime manifolds of the form <span>((0,t_0]times mathbb {T}{}^{n-1})</span>, <span>(t_0&gt;0)</span>. Stability is established under the assumption that the initial data is <i>synchronized</i>, which means that on the initial hypersurface <span>(Sigma = {t_0}times mathbb {T}{}^{n-1})</span>, the scalar field <span>(tau = exp bigl (sqrt{frac{2(n-2)}{n-1}}phi bigr ) )</span> is constant, that is, <span>(Sigma =tau ^{-1}({t_0}))</span>. As we show that all initial data sets that are sufficiently close to FRLW ones can be evolved via the Einstein-scalar field equation into new initial data sets that are <i>synchronized</i>, no generality is lost by this assumption. By using <span>(tau )</span> as a time coordinate, we establish that the perturbed FLRW spacetime manifolds are of the form <span>(M = bigcup _{tin (0,t_0]}tau ^{-1}({t})cong (0,t_0]times mathbb {T}{}^{n-1})</span>, the perturbed FLRW solutions are asymptotically pointwise Kasner as <span>(tau searrow 0)</span>, and a big bang singularity, characterised by the blow up of the scalar curvature, occurs at <span>(tau =0)</span>. An important aspect of our past stability proof is that we use a hyperbolic gauge reduction of the Einstein-scalar field equations. As a consequence, all of the estimates used in the stability proof can be localized and we employ this property to establish a corresponding localized past stability result for the FLRW solutions.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138559897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Relaxation Approximation and Asymptotic Stability of Stratified Solutions to the IPM Equation IPM 方程分层解的松弛逼近和渐近稳定性
IF 2.5 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2023-12-08 DOI: 10.1007/s00205-023-01945-x
Roberta Bianchini, Timothée Crin-Barat, Marius Paicu
{"title":"Relaxation Approximation and Asymptotic Stability of Stratified Solutions to the IPM Equation","authors":"Roberta Bianchini,&nbsp;Timothée Crin-Barat,&nbsp;Marius Paicu","doi":"10.1007/s00205-023-01945-x","DOIUrl":"10.1007/s00205-023-01945-x","url":null,"abstract":"<div><p>We prove the nonlinear asymptotic stability of stably stratified solutions to the Incompressible Porous Media equation (IPM) for initial perturbations in <span>(dot{H}^{1-tau }(mathbb {R}^2) cap dot{H}^s(mathbb {R}^2))</span> with <span>(s &gt; 3)</span> and for any <span>(0&lt; tau &lt;1)</span>. Such a result improves upon the existing literature, where the asymptotic stability is proved for initial perturbations belonging at least to <span>(H^{20}(mathbb {R}^2))</span>. More precisely, the aim of the article is threefold. First, we provide a simplified and improved proof of global-in-time well-posedness of the Boussinesq equations with strongly damped vorticity in <span>(H^{1-tau }(mathbb {R}^2) cap dot{H}^s(mathbb {R}^2))</span> with <span>(s &gt; 3)</span> and <span>(0&lt; tau &lt;1)</span>. Next, we prove the strong convergence of the Boussinesq system with damped vorticity towards (IPM) under a suitable scaling. Lastly, the asymptotic stability of stratified solutions to (IPM) follows as a byproduct. A symmetrization of the approximating system and a careful study of the anisotropic properties of the equations via anisotropic Littlewood-Paley decomposition play key roles to obtain uniform energy estimates. Finally, one of the main new and crucial points is the integrable time decay of the vertical velocity <span>(Vert u_2(t)Vert _{L^infty (mathbb {R}^2)})</span> for initial data only in <span>(dot{H}^{1-tau }(mathbb {R}^2) cap dot{H}^s(mathbb {R}^2))</span> with <span>(s &gt;3)</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138553007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Consistency of the Flat Flow Solution to the Volume Preserving Mean Curvature Flow 体积保持平均曲率流的平流解的一致性
IF 2.5 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2023-12-07 DOI: 10.1007/s00205-023-01944-y
Vesa Julin, Joonas Niinikoski
{"title":"Consistency of the Flat Flow Solution to the Volume Preserving Mean Curvature Flow","authors":"Vesa Julin,&nbsp;Joonas Niinikoski","doi":"10.1007/s00205-023-01944-y","DOIUrl":"10.1007/s00205-023-01944-y","url":null,"abstract":"<div><p>We consider the flat flow solution, obtained via a discrete minimizing movement scheme, to the volume preserving mean curvature flow starting from <span>(C^{1,1})</span>-regular set. We prove the consistency principle, which states that (any) flat flow solution agrees with the classical solution as long as the latter exists. In particular the flat flow solution is unique and smooth up to the first singular time. We obtain the result by proving the full regularity for the discrete time approximation of the flat flow such that the regularity estimates are stable with respect to the time discretization. Our method can also be applied in the case of the mean curvature flow and thus it provides an alternative proof, not relying on comparison principle, for the consistency between the flat flow solution and the classical solution for <span>(C^{1,1})</span>-regular initial sets.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-023-01944-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138552996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fine Properties of Geodesics and Geodesic (lambda )-Convexity for the Hellinger–Kantorovich Distance 测地线和测地线的精细性质(lambda ) - Hellinger-Kantorovich距离的凸性
IF 2.5 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2023-11-29 DOI: 10.1007/s00205-023-01941-1
Matthias Liero, Alexander Mielke, Giuseppe Savaré
{"title":"Fine Properties of Geodesics and Geodesic (lambda )-Convexity for the Hellinger–Kantorovich Distance","authors":"Matthias Liero,&nbsp;Alexander Mielke,&nbsp;Giuseppe Savaré","doi":"10.1007/s00205-023-01941-1","DOIUrl":"10.1007/s00205-023-01941-1","url":null,"abstract":"<div><p>We study the fine regularity properties of optimal potentials for the dual formulation of the Hellinger–Kantorovich problem (<span>(textsf{H}!!textsf{K})</span>), providing sufficient conditions for the solvability of the primal Monge formulation. We also establish new regularity properties for the solution of the Hamilton–Jacobi equation arising in the dual dynamic formulation of <span>(textsf{H}!!textsf{K})</span>, which are sufficiently strong to construct a characteristic transport-growth flow driving the geodesic interpolation between two arbitrary positive measures. These results are applied to study relevant geometric properties of <span>(textsf{H}!!textsf{K})</span> geodesics and to derive the convex behaviour of their Lebesgue density along the transport flow. Finally, exact conditions for functionals defined on the space of measures are derived that guarantee the geodesic <span>(lambda )</span>-convexity with respect to the Hellinger–Kantorovich distance. Examples of geodesically convex functionals are provided.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-023-01941-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138468335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Functions with Bounded Hessian–Schatten Variation: Density, Variational, and Extremality Properties 有界Hessian-Schatten变分函数:密度、变分和极值性质
IF 2.5 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2023-11-20 DOI: 10.1007/s00205-023-01938-w
Luigi Ambrosio, Camillo Brena, Sergio Conti
{"title":"Functions with Bounded Hessian–Schatten Variation: Density, Variational, and Extremality Properties","authors":"Luigi Ambrosio,&nbsp;Camillo Brena,&nbsp;Sergio Conti","doi":"10.1007/s00205-023-01938-w","DOIUrl":"10.1007/s00205-023-01938-w","url":null,"abstract":"<div><p>In this paper we analyze in detail a few questions related to the theory of functions with bounded <i>p</i>-Hessian–Schatten total variation, which are relevant in connection with the theory of inverse problems and machine learning. We prove an optimal density result, relative to the <i>p</i>-Hessian–Schatten total variation, of continuous piecewise linear (CPWL) functions in any space dimension <i>d</i>, using a construction based on a mesh whose local orientation is adapted to the function to be approximated. We show that not all extremal functions with respect to the <i>p</i>-Hessian–Schatten total variation are CPWL. Finally, we prove the existence of minimizers of certain relevant functionals involving the <i>p</i>-Hessian–Schatten total variation in the critical dimension <span>(d=2)</span>.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-023-01938-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138431519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Two-Dimensional Ferronematics, Canonical Harmonic Maps and Minimal Connections 二维铁线学，正则调和映射和最小连接
IF 2.5 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2023-11-17 DOI: 10.1007/s00205-023-01937-x
Giacomo Canevari, Apala Majumdar, Bianca Stroffolini, Yiwei Wang
{"title":"Two-Dimensional Ferronematics, Canonical Harmonic Maps and Minimal Connections","authors":"Giacomo Canevari,&nbsp;Apala Majumdar,&nbsp;Bianca Stroffolini,&nbsp;Yiwei Wang","doi":"10.1007/s00205-023-01937-x","DOIUrl":"10.1007/s00205-023-01937-x","url":null,"abstract":"<div><p>We study a variational model for ferronematics in two-dimensional domains, in the “super-dilute” regime. The free energy functional consists of a reduced Landau-de Gennes energy for the nematic order parameter, a Ginzburg–Landau type energy for the spontaneous magnetisation, and a coupling term that favours the co-alignment of the nematic director and the magnetisation. In a suitable asymptotic regime, we prove that the nematic order parameter converges to a canonical harmonic map with non-orientable point defects, while the magnetisation converges to a singular vector field, with line defects that connect the non-orientable point defects in pairs, along a minimal connection.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-023-01937-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138138483","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Weak and Strong Versions of the Kolmogorov 4/5-Law for Stochastic Burgers Equation 随机Burgers方程的Kolmogorov 4/5定律的弱和强版本
IF 2.5 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2023-11-12 DOI: 10.1007/s00205-023-01940-2
Peng Gao, Sergei Kuksin
{"title":"Weak and Strong Versions of the Kolmogorov 4/5-Law for Stochastic Burgers Equation","authors":"Peng Gao,&nbsp;Sergei Kuksin","doi":"10.1007/s00205-023-01940-2","DOIUrl":"10.1007/s00205-023-01940-2","url":null,"abstract":"<div><p>For solutions of the space-periodic stochastic 1d Burgers equation we establish two versions of the Kolmogorov 4/5-law; this provides an asymptotic expansion for the third moment of increments of turbulent velocity fields. We also prove for this equation an analogy of the Landau objection to possible universality of Kolmogorov’s theory of turbulence, and show that the third moment is the only one which admits a universal asymptotic expansion.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134795990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Free Boundary Minimal Annuli Immersed in the Unit Ball 自由边界最小环空浸入单位球
IF 2.5 1区 数学
Archive for Rational Mechanics and Analysis Pub Date : 2023-11-10 DOI: 10.1007/s00205-023-01943-z
Isabel Fernández, Laurent Hauswirth, Pablo Mira
{"title":"Free Boundary Minimal Annuli Immersed in the Unit Ball","authors":"Isabel Fernández,&nbsp;Laurent Hauswirth,&nbsp;Pablo Mira","doi":"10.1007/s00205-023-01943-z","DOIUrl":"10.1007/s00205-023-01943-z","url":null,"abstract":"<div><p>We construct a family of compact free boundary minimal annuli immersed in the unit ball <span>(mathbb {B}^3)</span> of <span>(mathbb {R}^3)</span>, the first such examples other than the critical catenoid. This solves a problem formulated by Nitsche in 1985. These annuli are symmetric with respect to two orthogonal planes and a finite group of rotations around an axis, and are foliated by spherical curvature lines. We show that the only free boundary minimal annulus embedded in <span>(mathbb {B}^3)</span> foliated by spherical curvature lines is the critical catenoid; in particular, the minimal annuli that we construct are not embedded. On the other hand, we also construct families of non-rotational compact embedded capillary minimal annuli in <span>(mathbb {B}^3)</span>. Their existence solves in the negative a problem proposed by Wente in 1995.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-023-01943-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134795974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
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