{"title":"Cylindrical singular minimal surfaces","authors":"Ulrich Dierkes, Rafael López","doi":"10.4171/rlm/1018","DOIUrl":"https://doi.org/10.4171/rlm/1018","url":null,"abstract":"We construct a foliation of the upper half space in $mathbb{R}^3$ consisting of minimizing cylindrical $alpha$-singular minimal surfaces when $alpha<0$. Furthermore, we discuss stability results for the $alpha$-catenaries when $alpha>0$ and relate these to the geodesics of a conformal metric.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136234397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"$*$-logarithm for slice regular functions","authors":"Amedeo Altavilla, Chiara de Fabritiis","doi":"10.4171/rlm/1016","DOIUrl":"https://doi.org/10.4171/rlm/1016","url":null,"abstract":"In this paper, we study the (possible) solutions of the equation $exp_(f)=g$, where $g$ is a slice regular never vanishing function on a circular domain of the quaternions $mathbb{H}$ and $exp_$ is the natural generalization of the usual exponential to the algebra of slice regular functions. Any function $f$ which satisfies $exp_(f)=g$ is called a $$-logarithm of $g$. We provide necessary and sufficient conditions, expressed in terms of the zero set of the “vector” part $g_v$ of $g$, for the existence of a $*$-logarithm of $g$, under a natural topological condition on the domain $Omega$. By this way, we prove an existence result if $g_v$ has no non-real isolated zeroes; we are also able to give a comprehensive approach to deal with more general cases. We are thus able to obtain an existence result when the non-real isolated zeroes of $g_v$ are finite, the domain is either the unit ball, or $mathbb{H}$, or $mathbb{D}$ (the solid torus obtained by circularization in $mathbb{H}$ of the disc contained in $mathbb{C}$ and centered in $2sqrt{-1}$ with radius $1$), and a further condition on the “real part” $g_0$ of $g$ is satisfied (see Theorem 6.19 for a precise statement). We also find some unexpected uniqueness results, again related to the zero set of $g_v$, in sharp contrast with the complex case. A number of examples are given throughout the paper in order to show the sharpness of the required conditions.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Two-phase models for elastic membranes with soft inclusions","authors":"Mario Santilli, Bernd Schmidt","doi":"10.4171/rlm/1012","DOIUrl":"https://doi.org/10.4171/rlm/1012","url":null,"abstract":"We derive an effective membrane theory in the thin film limit within a two-phase material model for a specimen consisting of an elastic matrix and soft inclusions or voids. These inclusions may lead to the formation of cracks within the elastic matrix and the corresponding limiting models are described by Griffith-type fracture energy functionals. We also provide simplified proofs of relaxation results for bulk materials.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136254448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Holomorphic functions on the disk with infinitely many zeros","authors":"Sebastiano Boscardin","doi":"10.4171/rlm/1015","DOIUrl":"https://doi.org/10.4171/rlm/1015","url":null,"abstract":"Suppose that $phi(z)=sum_{n=1}^infty d_nz^n$ is a series with radius of convergence greater than 1 and suppose that $d_n$ are real numbers such that $phi(1)neq0$. We prove that for any integer $a$ greater than $1$, the complex variable function $$ f(z)=sum_{n=0}^{infty}phi(z^{a^n})=phi(z)+phi(z^a)+phi(z^{a^2})+phi(z^{a^3})+cdots $$ has infinitely many zeros on the unit disk. It even takes every complex value in every disk centered in any point of the boundary.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136293501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Filomena De Filippis, Francesco Leonetti, Paolo Marcellini, Elvira Mascolo
{"title":"The Sobolev class where a weak solution is a local minimizer","authors":"Filomena De Filippis, Francesco Leonetti, Paolo Marcellini, Elvira Mascolo","doi":"10.4171/rlm/1014","DOIUrl":"https://doi.org/10.4171/rlm/1014","url":null,"abstract":"The aim of this paper is to propose some results which we hope could contribute to understand better Lavrentiev's phenomenon for energy integrals as in (1.1) under some $p,q$-growth conditions as in (1.2); in fact, we expect that Lavrentiev's phenomenon does not occur if the quotient $q/p$ is not too large in dependence of $n$, for instance, as in the cases – either scalar or vectorial ones – that we consider in this manuscript.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136292363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Overdetermined problems and relative Cheeger sets in unbounded domains","authors":"Danilo Gregorin Afonso, Alessandro Iacopetti, Filomena Pacella","doi":"10.4171/rlm/1017","DOIUrl":"https://doi.org/10.4171/rlm/1017","url":null,"abstract":"In this paper, we study a partially overdetermined mixed boundary value problem for domains $Omega$ contained in an unbounded set $mathcal{C}$. We introduce the notion of the Cheeger set relative to $mathcal{C}$ and show that if a domain $Omegasubsetmathcal{C}$ admits a solution of the overdetermined problem, then it coincides with its relative Cheeger set. We also study the related problem of characterizing constant mean curvature surfaces $Gamma$ inside $mathcal{C}$. In the case when $mathcal{C}$ is a cylinder, we obtain further results whenever the relative boundary of $Omega$ or the surface $Gamma$ is a graph on the base of the cylinder.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135345969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fano's last Fano","authors":"Marco Andreatta, Roberto Pignatelli","doi":"10.4171/rlm/1010","DOIUrl":"https://doi.org/10.4171/rlm/1010","url":null,"abstract":"In 1949, Fano published his last paper on $3$-folds with canonical sectional curves. There, he constructed and described a $3$-fold of degree $22$ in a projective space of dimension $13$ with canonical curve section, which we like to call Fano's last Fano. We report on Fano's construction and we provide various (in our opinion missing) proofs in modern language. Moreover, we try to use results and techniques available at that time. After that we construct Fano's last Fano with modern tools, in particular, via the Hilbert scheme of zero cycles on a rational surface; as a consequence we easily point out the corresponding example in the Mori–Mukai classification.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135345971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on some non-local variational problems","authors":"Davide Carazzato","doi":"10.4171/rlm/1006","DOIUrl":"https://doi.org/10.4171/rlm/1006","url":null,"abstract":"We study two non-local variational problems that are characterized by the presence of a Riesz-like repulsive term that competes with an attractive term. The first functional is defined on the subsets of the Euclidean space and has the fractional perimeter as an attractive term. The second functional instead is defined on non-negative integrable and uniformly bounded densities and contains an attractive term of positive-power type. For both of the functionals, we prove that balls are the unique minimizers in the appropriate volume constraint range, generalizing the results already present in the literature for more specific energies.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A counterexample to the monotone increasing behavior of an Alt–Caffarelli–Friedman formula in the Heisenberg group","authors":"Fausto Ferrari, Nicolò Forcillo","doi":"10.4171/rlm/1007","DOIUrl":"https://doi.org/10.4171/rlm/1007","url":null,"abstract":"In this paper, we provide a counterexample about the existence of an increasing monotonicity behavior of a function introduced by Ferrari and Forcillo (2020), companion of the celebrated Alt–Caffarelli–Friedman monotonicity formula, in the noncommutative framework.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134944567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Regularity improvement for the minimizers of the two-dimensional Griffith energy","authors":"Camille Labourie, Antoine Lemenant","doi":"10.4171/rlm/1009","DOIUrl":"https://doi.org/10.4171/rlm/1009","url":null,"abstract":"In this paper, we prove that the singular set of connected minimizers of the planar Griffith functional has Hausdorff dimension strictly less than one, together with the higher integrability of the symmetrized gradient.","PeriodicalId":80594,"journal":{"name":"Atti della Accademia nazionale dei Lincei","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135302006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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