M. A. D. de Cataldo, D. Maulik, Junliang Shen, Siqing Zhang
{"title":"Cohomology of the moduli of Higgs bundles on a curve via positive characteristic","authors":"M. A. D. de Cataldo, D. Maulik, Junliang Shen, Siqing Zhang","doi":"10.4171/jems/1393","DOIUrl":"https://doi.org/10.4171/jems/1393","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138967703","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}
. In this paper, we obtain sharp estimates for the rate of propagation of the Fisher-KPP equation with nonlocal diffusion and free boundaries. The nonlocal diffusion operator is given by (cid:82) R J ( x − y ) u ( t, y ) dy − u ( t, x ), and our estimates hold for some typical classes of kernel functions J ( x ). For example, if for | x | (cid:29) 1 the kernel function satisfies J ( x ) ∼ | x | − γ with γ > 1, then it follows from [17] that there is a finite spreading speed when γ > 2, namely the free boundary x = h ( t ) satisfies lim t →∞ h ( t ) /t = c 0 for some uniquely determined positive constant c 0 depending on J , and when γ ∈ (1 , 2], lim t →∞ h ( t ) /t = ∞ ; the estimates in the current paper imply that, for t (cid:29) 1, c 0 t − h ( t ) ∼ 1 when γ > 3 ln t when γ = 3 , t 3 − γ when γ ∈ (2 , 3) , and h ( t ) ∼ (cid:26) t ln t when γ = 2 , t 1 / ( γ − 1) when γ ∈ (1 , 2) . Our approach is based on subtle integral estimates and constructions of upper and lower solutions, which rely crucially on guessing correctly the order of growth of the term to be estimated. The techniques developed here lay the ground for extensions to more general situations.
.在本文中，我们得到了具有非局部二重扩散和自由边界的 Fisher-KPP 方程传播速度的精确估计值。非局部扩散算子由 (cid:82) R J ( x - y ) u ( t, y ) dy - u ( t, x ) 给出，我们的估计值对于一些典型的核函数 J ( x ) 类是成立的。例如，如果对于 | x | (cid：29) 1 的核函数满足 J ( x ) ∼ | x | - γ，且 γ > 1，那么根据[17]，当 γ > 2 时，存在一个有限的扩散速度、即自由边界 x = h ( t ) 对于某个取决于 J 的唯一确定的正常数 c 0 满足 lim t →∞ h ( t ) /t = c 0，而当γ∈（1 , 2]时，lim t →∞ h ( t ) /t = ∞；本文的估计意味着，对于 t （cid：29) 1 时，当 γ > 3 时，c 0 t - h ( t ) ∼ 1 ；当 γ = 3 时，t 3 - γ ；当 γ ∈ (2 , 3) 时，h ( t ) ∼ (cid:26) t ；当 γ = 2 时，t 1 / ( γ - 1) ；当 γ ∈ (1 , 2) 时，h ( t ) ∼ (cid:26) t 。我们的方法基于微妙的积分估计和上下限解的构造，其关键在于正确猜测待估计项的增长阶数。这里开发的技术为扩展到更一般的情况奠定了基础。
{"title":"Rate of propagation for the Fisher-KPP equation with nonlocal diffusion and free boundaries","authors":"Yihong Du, W. Ni","doi":"10.4171/jems/1392","DOIUrl":"https://doi.org/10.4171/jems/1392","url":null,"abstract":". In this paper, we obtain sharp estimates for the rate of propagation of the Fisher-KPP equation with nonlocal diffusion and free boundaries. The nonlocal diffusion operator is given by (cid:82) R J ( x − y ) u ( t, y ) dy − u ( t, x ), and our estimates hold for some typical classes of kernel functions J ( x ). For example, if for | x | (cid:29) 1 the kernel function satisfies J ( x ) ∼ | x | − γ with γ > 1, then it follows from [17] that there is a finite spreading speed when γ > 2, namely the free boundary x = h ( t ) satisfies lim t →∞ h ( t ) /t = c 0 for some uniquely determined positive constant c 0 depending on J , and when γ ∈ (1 , 2], lim t →∞ h ( t ) /t = ∞ ; the estimates in the current paper imply that, for t (cid:29) 1, c 0 t − h ( t ) ∼ 1 when γ > 3 ln t when γ = 3 , t 3 − γ when γ ∈ (2 , 3) , and h ( t ) ∼ (cid:26) t ln t when γ = 2 , t 1 / ( γ − 1) when γ ∈ (1 , 2) . Our approach is based on subtle integral estimates and constructions of upper and lower solutions, which rely crucially on guessing correctly the order of growth of the term to be estimated. The techniques developed here lay the ground for extensions to more general situations.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":2.6,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138995829","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}
Eli Glasner, Wen Huang, Song Shao, Benjamin Weiss, Xiangdong Ye
We prove that the maximal infinite step pro-nilfactor $X_infty$ of a minimal dynamical system $(X,T)$ is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of $pi:X rightarrow X_infty$, the induced open extension $pi^*:X^* rightarrow X^*_infty$ has the following property: for $x$ in a dense $G_delta$ set of $X^*$, the orbit closure $L_x=overline{mathcal{O}}((x,x,ldots,x), Ttimes T^2times ldots times T^d)$ is $(pi^*)^{(d)}$-saturated, i.e. $L_x=((pi^*)^{(d)})^{-1}(pi^*)^{(d)}(L_x)$.
{"title":"Topological characteristic factors and nilsystems","authors":"Eli Glasner, Wen Huang, Song Shao, Benjamin Weiss, Xiangdong Ye","doi":"10.4171/jems/1379","DOIUrl":"https://doi.org/10.4171/jems/1379","url":null,"abstract":"We prove that the maximal infinite step pro-nilfactor $X_infty$ of a minimal dynamical system $(X,T)$ is the topological characteristic factor in a certain sense. Namely, we show that by an almost one to one modification of $pi:X rightarrow X_infty$, the induced open extension $pi^*:X^* rightarrow X^*_infty$ has the following property: for $x$ in a dense $G_delta$ set of $X^*$, the orbit closure $L_x=overline{mathcal{O}}((x,x,ldots,x), Ttimes T^2times ldots times T^d)$ is $(pi^*)^{(d)}$-saturated, i.e. $L_x=((pi^*)^{(d)})^{-1}(pi^*)^{(d)}(L_x)$. ","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135870959","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}
Jason Bell, Keping Huang, Wayne Peng, Thomas Tucker
We prove an analog of the Tits alternative for endomorphisms of $mathbb{P}^1$. In particular, we show that if $S$ is a finitely generated semigroup of endomorphisms of $mathbb{P}^1$ over $mathbb{C}$, then either $S$ has polynomially bounded growth or $S$ contains a nonabelian free semigroup. We also show that if $f$ and $g$ are polarizable maps over any field of any characteristic and $operatorname{Prep}(f) not= operatorname{Prep}(g)$, then for all sufficiently large $j$, the semigroup $langle f^j, g^j rangle$ is a free semigroup on two generators.
{"title":"A Tits alternative for endomorphisms of the projective line","authors":"Jason Bell, Keping Huang, Wayne Peng, Thomas Tucker","doi":"10.4171/jems/1376","DOIUrl":"https://doi.org/10.4171/jems/1376","url":null,"abstract":"We prove an analog of the Tits alternative for endomorphisms of $mathbb{P}^1$. In particular, we show that if $S$ is a finitely generated semigroup of endomorphisms of $mathbb{P}^1$ over $mathbb{C}$, then either $S$ has polynomially bounded growth or $S$ contains a nonabelian free semigroup. We also show that if $f$ and $g$ are polarizable maps over any field of any characteristic and $operatorname{Prep}(f) not= operatorname{Prep}(g)$, then for all sufficiently large $j$, the semigroup $langle f^j, g^j rangle$ is a free semigroup on two generators.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135808401","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}
We prove a sharp stability result for the Brunn–Minkowski inequality for $A,Bsubsetmathbb{R}^2$. Assuming that the Brunn–Minkowski deficit $delta=|A+B|^{1/2}/(|A|^{1/2}+|B|^{1/2})-1$ is sufficiently small in terms of $t=|A|^{1/2}/(|A|^{1/2}+|B|^{1/2})$, there exist homothetic convex sets $K_A supset A$ and $K_Bsupset B$ such that $frac{|K_Asetminus A|}{|A|}+frac{|K_Bsetminus B|}{|B|} le C t^{-{1/2}}delta^{1/2}$. The key ingredient is to show for every $epsilon,t>0$, if $delta$ is sufficiently small then $|!operatorname{co}(A+B)setminus (A+B)|le (1+epsilon)(|!operatorname{co}(A)setminus A|+|!operatorname{co}(B)setminus B|)$.
对于$A,Bsubsetmathbb{R}^2$，我们证明了Brunn-Minkowski不等式的一个尖锐的稳定性结果。假设Brunn-Minkowski赤字$delta=|A+B|^{1/2}/(|A|^{1/2}+|B|^{1/2})-1$在$t=|A|^{1/2}/(|A|^{1/2}+|B|^{1/2})$上足够小，则存在齐次凸集$K_A supset A$和$K_Bsupset B$，使得$frac{|K_Asetminus A|}{|A|}+frac{|K_Bsetminus B|}{|B|} le C t^{-{1/2}}delta^{1/2}$。关键是要显示对于每个$epsilon,t>0$，如果$delta$足够小，那么$|!operatorname{co}(A+B)setminus (A+B)|le (1+epsilon)(|!operatorname{co}(A)setminus A|+|!operatorname{co}(B)setminus B|)$。
{"title":"Sharp quantitative stability of the planar Brunn–Minkowski inequality","authors":"Peter van Hintum, Hunter Spink, Marius Tiba","doi":"10.4171/jems/1372","DOIUrl":"https://doi.org/10.4171/jems/1372","url":null,"abstract":"We prove a sharp stability result for the Brunn–Minkowski inequality for $A,Bsubsetmathbb{R}^2$. Assuming that the Brunn–Minkowski deficit $delta=|A+B|^{1/2}/(|A|^{1/2}+|B|^{1/2})-1$ is sufficiently small in terms of $t=|A|^{1/2}/(|A|^{1/2}+|B|^{1/2})$, there exist homothetic convex sets $K_A supset A$ and $K_Bsupset B$ such that $frac{|K_Asetminus A|}{|A|}+frac{|K_Bsetminus B|}{|B|} le C t^{-{1/2}}delta^{1/2}$. The key ingredient is to show for every $epsilon,t>0$, if $delta$ is sufficiently small then $|!operatorname{co}(A+B)setminus (A+B)|le (1+epsilon)(|!operatorname{co}(A)setminus A|+|!operatorname{co}(B)setminus B|)$.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134972736","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}
We establish a comparison isomorphism between prismatic cohomology and derived de Rham cohomology respecting various structures, such as their Frobenius actions and filtrations. As an application, when $X$ is a proper smooth formal scheme over $mathcal O_K$ with $K$ being a $p$-adic field, we improve Breuil--Caruso's theory on comparison between torsion crystalline cohomology and torsion 'etale cohomology.
{"title":"Comparison of prismatic cohomology and derived de Rham cohomology","authors":"Shizhang Li, Tong Liu","doi":"10.4171/jems/1377","DOIUrl":"https://doi.org/10.4171/jems/1377","url":null,"abstract":"We establish a comparison isomorphism between prismatic cohomology and derived de Rham cohomology respecting various structures, such as their Frobenius actions and filtrations. As an application, when $X$ is a proper smooth formal scheme over $mathcal O_K$ with $K$ being a $p$-adic field, we improve Breuil--Caruso's theory on comparison between torsion crystalline cohomology and torsion 'etale cohomology.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135217936","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}
Braverman and Kazhdan proposed a conjecture, later refined by Ngo and broadened to the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson summation formulae. The first author in joint work with B.~Liu and later the first two authors proved these conjectures for certain spherical varieties $Y$ built out of triples of quadratic spaces. However, in these works the Fourier transform was only proven to exist. In the present paper we give, for the first time, an explicit formula for the Fourier transform on $Y.$ We also prove that it is unitary in the nonarchimedean case. As preparation for this result, we give explicit formulae for Fourier transforms on Braverman-Kazhdan spaces attached to maximal parabolic subgroups of split, simple, simply connected groups. These Fourier transforms are of independent interest, for example, from the point of view of analytic number theory.
{"title":"Harmonic analysis on certain spherical varieties","authors":"Jayce R. Getz, Chun-Hsien Hsu, Spencer Leslie","doi":"10.4171/jems/1381","DOIUrl":"https://doi.org/10.4171/jems/1381","url":null,"abstract":"Braverman and Kazhdan proposed a conjecture, later refined by Ngo and broadened to the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson summation formulae. The first author in joint work with B.~Liu and later the first two authors proved these conjectures for certain spherical varieties $Y$ built out of triples of quadratic spaces. However, in these works the Fourier transform was only proven to exist. In the present paper we give, for the first time, an explicit formula for the Fourier transform on $Y.$ We also prove that it is unitary in the nonarchimedean case. As preparation for this result, we give explicit formulae for Fourier transforms on Braverman-Kazhdan spaces attached to maximal parabolic subgroups of split, simple, simply connected groups. These Fourier transforms are of independent interest, for example, from the point of view of analytic number theory.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135323065","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}
Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The degree of nonminimality is the minimum number of realisations of the type required to witness a nonalgebraic forking extension. Conditional on the truth of a conjecture of Borovik and Cherlin on the generic multiple-transitivity of homogeneous spaces definable in the stable theory being considered, it is shown that the nonminimality degree is bounded by the $U$-rank plus 2. The Borovik–Cherlin conjecture itself is verified for algebraic and meromorphic group actions, and a bound of $U$-rank plus 1 is then deduced unconditionally for differentially closed fields and compact complex manifolds. An application is given regarding transcendence of solutions to algebraic differential equations.
{"title":"Bounding nonminimality and a conjecture of Borovik–Cherlin","authors":"James Freitag, Rahim Moosa","doi":"10.4171/jems/1384","DOIUrl":"https://doi.org/10.4171/jems/1384","url":null,"abstract":"Motivated by the search for methods to establish strong minimality of certain low order algebraic differential equations, a measure of how far a finite rank stationary type is from being minimal is introduced and studied: The degree of nonminimality is the minimum number of realisations of the type required to witness a nonalgebraic forking extension. Conditional on the truth of a conjecture of Borovik and Cherlin on the generic multiple-transitivity of homogeneous spaces definable in the stable theory being considered, it is shown that the nonminimality degree is bounded by the $U$-rank plus 2. The Borovik–Cherlin conjecture itself is verified for algebraic and meromorphic group actions, and a bound of $U$-rank plus 1 is then deduced unconditionally for differentially closed fields and compact complex manifolds. An application is given regarding transcendence of solutions to algebraic differential equations.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136057755","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}
We interpret the log-Brunn–Minkowski conjecture of Böröczky–Lutwak–Yang–Zhang as a spectral problem in centro-affine differential geometry. In particular, we show that the Hilbert--Brunn--Minkowski operator coincides with the centro-affine Laplacian, thus obtaining a new avenue for tackling the conjecture using insights from affine differential geometry. As every strongly convex hypersurface in $mathbb{R}^n$ is a centro-affine unit sphere, it has constant centro-affine Ricci curvature equal to $n-2$, in stark contrast to the standard weighted Ricci curvature of the associated metric-measure space, which will in general be negative. In particular, we may use the classical argument of Lichnerowicz and a centro-affine Bochner formula to give a new proof of the Brunn–Minkowski inequality. For origin-symmetric convex bodies enjoying fairly generous curvature pinching bounds (improving with dimension), we are able to show global uniqueness in the $L^p$- and log-Minkowski problems, as well as the corresponding global $L^p$- and log-Minkowski conjectured inequalities. As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body $bar K$ in $mathbb{R}^n$, there exists an origin-symmetric convex body $K$ with $bar K subset K subset 8 bar K$ such that $K$ satisfies the log-Minkowski conjectured inequality, and such that $K$ is uniquely determined by its cone-volume measure $V_K$. If $bar K$ is not extremely far from a Euclidean ball to begin with, an analogous isometric result, where $8$ is replaced by $1+varepsilon$, is obtained as well.
我们将Böröczky-Lutwak-Yang-Zhang的log-Brunn-Minkowski猜想解释为中心仿射微分几何中的一个谱问题。特别是，我们证明了Hilbert- Brunn- Minkowski算子与中心仿射拉普拉斯算子重合，从而获得了利用仿射微分几何的见解来解决猜想的新途径。由于$mathbb{R}^n$中的每个强凸超曲面都是中心仿射单位球，因此它具有恒定的中心仿射Ricci曲率，等于$n-2$，与相关度量度量空间的标准加权Ricci曲率形成鲜明对比，后者通常为负。特别地，我们可以利用Lichnerowicz的经典论证和中心仿射Bochner公式给出布伦-闵可夫斯基不等式的新证明。对于原点对称凸体具有相当大的曲率挤压界(随维数的增加而提高)，我们能够在$L^p$ -和log-Minkowski问题中显示全局唯一性，以及相应的全局$L^p$ -和log-Minkowski猜想不等式。因此，我们解决了对数-闵可夫斯基问题的同构版本:对于$mathbb{R}^n$中的任意原点对称凸体$bar K$，存在一个具有$bar K subset K subset 8 bar K$的原点对称凸体$K$，使得$K$满足对数-闵可夫斯基猜想不等式，并且使得$K$是由其锥体积测度$V_K$唯一决定的。如果$bar K$一开始离欧几里得球不远，也可以得到类似的等距结果，其中$8$用$1+varepsilon$代替。
{"title":"Centro-affine differential geometry and the log-Minkowski problem","authors":"Emanuel Milman","doi":"10.4171/jems/1386","DOIUrl":"https://doi.org/10.4171/jems/1386","url":null,"abstract":"We interpret the log-Brunn–Minkowski conjecture of Böröczky–Lutwak–Yang–Zhang as a spectral problem in centro-affine differential geometry. In particular, we show that the Hilbert--Brunn--Minkowski operator coincides with the centro-affine Laplacian, thus obtaining a new avenue for tackling the conjecture using insights from affine differential geometry. As every strongly convex hypersurface in $mathbb{R}^n$ is a centro-affine unit sphere, it has constant centro-affine Ricci curvature equal to $n-2$, in stark contrast to the standard weighted Ricci curvature of the associated metric-measure space, which will in general be negative. In particular, we may use the classical argument of Lichnerowicz and a centro-affine Bochner formula to give a new proof of the Brunn–Minkowski inequality. For origin-symmetric convex bodies enjoying fairly generous curvature pinching bounds (improving with dimension), we are able to show global uniqueness in the $L^p$- and log-Minkowski problems, as well as the corresponding global $L^p$- and log-Minkowski conjectured inequalities. As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body $bar K$ in $mathbb{R}^n$, there exists an origin-symmetric convex body $K$ with $bar K subset K subset 8 bar K$ such that $K$ satisfies the log-Minkowski conjectured inequality, and such that $K$ is uniquely determined by its cone-volume measure $V_K$. If $bar K$ is not extremely far from a Euclidean ball to begin with, an analogous isometric result, where $8$ is replaced by $1+varepsilon$, is obtained as well.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254446","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}
In this paper we consider nonlinearly elastic, frame-indifferent, and singularly perturbed two-well models for materials undergoing solid-solid phase transitions in any space dimensions, and we perform a simultaneous passage to sharp-interface and small-strain limits. Sequences of deformations with equibounded energies are decomposed via suitable Caccioppoli partitions into the sum of piecewise constant rigid movements and suitably rescaled displacements. These converge to limiting partitions, deformations, and displacements, respectively. Whereas limiting deformations are simple laminates whose gradients only attain two values, the limiting displacements belong to the class of special functions with bounded variation (SBV). The latter feature elastic contributions measuring the distance to simple laminates, as well as jumps associated to two consecutive phase transitions having vanishing distance, and thus undetected by the limiting deformations. By $Gamma$- convergence we identify an effective limiting model given by the sum of a quadratic linearized elastic energy in terms of displacements along with two surface terms. The first one is proportional to the total length of interfaces created by jumps in the gradient of the limiting deformation. The second one is proportional to twice the total length of interfaces created by jumps in the limiting displacement, as well as by the boundaries of limiting partitions. A main tool of our analysis is a novel two-well rigidity estimate which has been derived in [Calc. Var. Partial Differential Equations 59, art. 44 (2020)] for a model with anisotropic second-order perturbation.
{"title":"Two-well linearization for solid-solid phase transitions","authors":"Elisa Davoli, Manuel Friedrich","doi":"10.4171/jems/1385","DOIUrl":"https://doi.org/10.4171/jems/1385","url":null,"abstract":"In this paper we consider nonlinearly elastic, frame-indifferent, and singularly perturbed two-well models for materials undergoing solid-solid phase transitions in any space dimensions, and we perform a simultaneous passage to sharp-interface and small-strain limits. Sequences of deformations with equibounded energies are decomposed via suitable Caccioppoli partitions into the sum of piecewise constant rigid movements and suitably rescaled displacements. These converge to limiting partitions, deformations, and displacements, respectively. Whereas limiting deformations are simple laminates whose gradients only attain two values, the limiting displacements belong to the class of special functions with bounded variation (SBV). The latter feature elastic contributions measuring the distance to simple laminates, as well as jumps associated to two consecutive phase transitions having vanishing distance, and thus undetected by the limiting deformations. By $Gamma$- convergence we identify an effective limiting model given by the sum of a quadratic linearized elastic energy in terms of displacements along with two surface terms. The first one is proportional to the total length of interfaces created by jumps in the gradient of the limiting deformation. The second one is proportional to twice the total length of interfaces created by jumps in the limiting displacement, as well as by the boundaries of limiting partitions. A main tool of our analysis is a novel two-well rigidity estimate which has been derived in [Calc. Var. Partial Differential Equations 59, art. 44 (2020)] for a model with anisotropic second-order perturbation.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254444","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}
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