We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of the arrangement.
{"title":"A differential algebra and the homotopy type of the complement of a toric arrangement","authors":"C. Concini, G. Gaiffi","doi":"10.4171/rlm/924","DOIUrl":"https://doi.org/10.4171/rlm/924","url":null,"abstract":"We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of the arrangement.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49340541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
D. Apushkinskaya, A. Nazarov, D. Palagachev, L. Softova
{"title":"Elliptic Venttsel problems with $VMO$ coefficients","authors":"D. Apushkinskaya, A. Nazarov, D. Palagachev, L. Softova","doi":"10.4171/RLM/896","DOIUrl":"https://doi.org/10.4171/RLM/896","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"15 1","pages":"391-399"},"PeriodicalIF":0.5,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81748491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quantitative estimates for sampling type operators with respect to the Jordan variation","authors":"L. Angeloni, D. Costarelli, G. Vinti","doi":"10.4171/rlm/890","DOIUrl":"https://doi.org/10.4171/rlm/890","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"12 1","pages":"269-284"},"PeriodicalIF":0.5,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83640796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"$H=W$ Musielak spaces framework","authors":"Youssef Ahmida, A. Fiorenza, A. Youssfi","doi":"10.4171/rlm/899","DOIUrl":"https://doi.org/10.4171/rlm/899","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"7 1","pages":"447-464"},"PeriodicalIF":0.5,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78713301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On relationship between H-distributions and microlocal compactness forms","authors":"N. Antonić, D. Mitrovic, L. Palle","doi":"10.4171/rlm/892","DOIUrl":"https://doi.org/10.4171/rlm/892","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"7 1","pages":"297-318"},"PeriodicalIF":0.5,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89187445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A model of capillary phenomena in RN with subcritical growth","authors":"C. Vetro","doi":"10.4171/rlm/894","DOIUrl":"https://doi.org/10.4171/rlm/894","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"78 1","pages":"335-347"},"PeriodicalIF":0.5,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80856240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Since the pioneering papers by E. Gagliardo and L. Nirenberg in 1959 the so called Sobolev-GagliardoNirenberg inequalities have not ceased to be one of the most useful resources for the mathematical treatment of linear and non-linear partial differential equations ([1], [3], [4], [7], [8], [10], [11], [16], [18], [19], [21], [23], [24] and [25], among an almost infinite list of references). This central role was increased with the more recent consideration of many nonlocal problems requiring non integer regularity exponents (see. e.g., the survey [20] and its many references). As usual, interpolation inequalities are obtained in a parallel way to the study of general embedding results on Sobolev spaces W (R) for different values of the exponents (for a very complete study of range of the valid exponents see [5]). The main goal of this paper is to revisit Sobolev type inequalities involving the fractional norm. It is known that the general embedding for the spaces W (R) can be obtained by interpolation theorems through the Besov space, see e.g., [1], [23], [17], [2], [21], [12], [13], and references therein. Here, we provide the proofs of the homogeneous fractional Sobolev Ẇ (R) embeddings and the trace results. Although the results below are known, our proof are self-contained and it seems to be novel by using the technique of the Hardy-Littlewood maximal functions and the sharp maximal function (see. e.g. [22] and [15]). Then, our results are as follows.
{"title":"Fractional Sobolev inequalities revisited: the maximal function approach","authors":"N. Dao, J. I. Díaz, Quoc-Hung Nguyen","doi":"10.4171/RLM/887","DOIUrl":"https://doi.org/10.4171/RLM/887","url":null,"abstract":"Since the pioneering papers by E. Gagliardo and L. Nirenberg in 1959 the so called Sobolev-GagliardoNirenberg inequalities have not ceased to be one of the most useful resources for the mathematical treatment of linear and non-linear partial differential equations ([1], [3], [4], [7], [8], [10], [11], [16], [18], [19], [21], [23], [24] and [25], among an almost infinite list of references). This central role was increased with the more recent consideration of many nonlocal problems requiring non integer regularity exponents (see. e.g., the survey [20] and its many references). As usual, interpolation inequalities are obtained in a parallel way to the study of general embedding results on Sobolev spaces W (R) for different values of the exponents (for a very complete study of range of the valid exponents see [5]). The main goal of this paper is to revisit Sobolev type inequalities involving the fractional norm. It is known that the general embedding for the spaces W (R) can be obtained by interpolation theorems through the Besov space, see e.g., [1], [23], [17], [2], [21], [12], [13], and references therein. Here, we provide the proofs of the homogeneous fractional Sobolev Ẇ (R) embeddings and the trace results. Although the results below are known, our proof are self-contained and it seems to be novel by using the technique of the Hardy-Littlewood maximal functions and the sharp maximal function (see. e.g. [22] and [15]). Then, our results are as follows.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"1 1","pages":"225-236"},"PeriodicalIF":0.5,"publicationDate":"2020-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82970699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We prove Lipschitz continuity results for solutions to a class of obstacle problems under standard growth conditions of p − type, p ≥ 2. The main novelty is the use of a linearization technique going back to [28] in order to interpret our constrained minimizer as a solution to a nonlinear elliptic equation, with a bounded right hand side. This lead us to start a Moser iteration scheme which provides the L ∞ bound for the gradient. The application of a recent higher differentiability result [24] allows us to simplify the procedure of the identification of the Radon measure in the linearization technique employed in [32]. To our knowdledge, this is the first result for non-automonous functionals with standard growth conditions in the direction of the Lipschitz regularity.
{"title":"Lipschitz continuity results for a class of obstacle problems","authors":"Carlo Benassi, Michele Caselli","doi":"10.4171/rlm/885","DOIUrl":"https://doi.org/10.4171/rlm/885","url":null,"abstract":". We prove Lipschitz continuity results for solutions to a class of obstacle problems under standard growth conditions of p − type, p ≥ 2. The main novelty is the use of a linearization technique going back to [28] in order to interpret our constrained minimizer as a solution to a nonlinear elliptic equation, with a bounded right hand side. This lead us to start a Moser iteration scheme which provides the L ∞ bound for the gradient. The application of a recent higher differentiability result [24] allows us to simplify the procedure of the identification of the Radon measure in the linearization technique employed in [32]. To our knowdledge, this is the first result for non-automonous functionals with standard growth conditions in the direction of the Lipschitz regularity.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2020-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/rlm/885","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43846902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Complex eigenvalue bounds for a Schrödinger operator on the half line","authors":"Francesco Ferrulli, A. Laptev","doi":"10.4171/rlm/876","DOIUrl":"https://doi.org/10.4171/rlm/876","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"226 1","pages":"1-13"},"PeriodicalIF":0.5,"publicationDate":"2020-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74980692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Junction of quasi-stationary ferromagnetic wires","authors":"K. Chacouche, L. Faella, C. Perugia","doi":"10.4171/rlm/878","DOIUrl":"https://doi.org/10.4171/rlm/878","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"2 1","pages":"25-56"},"PeriodicalIF":0.5,"publicationDate":"2020-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81750880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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