This paper is devoted to the determination of the cases where there is equality in Courant's nodal domain theorem in the case of a Robin boundary condition. For the square, we partially extend the results that were obtained by Pleijel, B'erard--Helffer, Helffer--Persson--Sundqvist for the Dirichlet and Neumann problems. After proving some general results that hold for any value of the Robin parameter $h$, we focus on the case when $h$ is large. We hope to come back to the analysis when $h$ is small in a second paper. We also obtain some semi-stability results for the number of nodal domains of a Robin eigenfunction of a domain with $C^{2,alpha}$ boundary ($alpha >0$) as $h$ large varies.
{"title":"Courant-sharp Robin eigenvalues for the square and other planar domains","authors":"K. Gittins, B. Helffer","doi":"10.4171/pm/2027","DOIUrl":"https://doi.org/10.4171/pm/2027","url":null,"abstract":"This paper is devoted to the determination of the cases where there is equality in Courant's nodal domain theorem in the case of a Robin boundary condition. For the square, we partially extend the results that were obtained by Pleijel, B'erard--Helffer, Helffer--Persson--Sundqvist for the Dirichlet and Neumann problems. \u0000After proving some general results that hold for any value of the Robin parameter $h$, we focus on the case when $h$ is large. We hope to come back to the analysis when $h$ is small in a second paper. \u0000We also obtain some semi-stability results for the number of nodal domains of a Robin eigenfunction of a domain with $C^{2,alpha}$ boundary ($alpha >0$) as $h$ large varies.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2027","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49625568","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 flows generated by vector fields with compact support","authors":"O. Kneuss, W. Neves","doi":"10.4171/PM/2013","DOIUrl":"https://doi.org/10.4171/PM/2013","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/PM/2013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45008722","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 revisit the Bohnenblust--Hille multilinear and polynomial inequalities and prove some new properties. Our main result is a multilinear version of a recent result on polynomials whose monomials have a uniformly bounded number of variables.
{"title":"Remarks on the Bohnenblust–Hille inequalities","authors":"D. Paulino, D. Pellegrino, Joedson Santos","doi":"10.4171/pm/2040","DOIUrl":"https://doi.org/10.4171/pm/2040","url":null,"abstract":"We revisit the Bohnenblust--Hille multilinear and polynomial inequalities and prove some new properties. Our main result is a multilinear version of a recent result on polynomials whose monomials have a uniformly bounded number of variables.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2040","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44254283","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 consider the scalar second order ODE u + |u | $alpha$ u + |u| $beta$ u = 0, where $alpha$, $beta$ are two positive numbers and the non-linear semi-group S(t) generated on IR 2 by the system in (u, u). We prove that S(t)IR 2 is bounded for all t > 0 whenever 0 0, u (t) 2 + |u(t)| $beta$+2 $le$ C max{t -- 2 $alpha$ , t -- ($alpha$+1)($beta$+2) $beta$--$alpha$ }.
我们考虑标量二阶ODE u + |u| $alpha$ u + |u| $beta$ u = 0，其中$alpha$, $beta$是两个正数，以及系统在(u, u)中在IR 2上生成的非线性半群S(t)。我们证明了S(t)IR 2对于所有t > 0都是有界的，当0 0时，u(t) 2 + |u(t)| $beta$ +2 $le$ C {maxt—2$alpha$, t—($alpha$ +1)($beta$ +2) $beta$—$alpha$。}
{"title":"The universal bound property for a class of second order ODEs","authors":"M. Abdelli, A. Haraux","doi":"10.4171/pm/2026","DOIUrl":"https://doi.org/10.4171/pm/2026","url":null,"abstract":"We consider the scalar second order ODE u + |u | $alpha$ u + |u| $beta$ u = 0, where $alpha$, $beta$ are two positive numbers and the non-linear semi-group S(t) generated on IR 2 by the system in (u, u). We prove that S(t)IR 2 is bounded for all t > 0 whenever 0 0, u (t) 2 + |u(t)| $beta$+2 $le$ C max{t -- 2 $alpha$ , t -- ($alpha$+1)($beta$+2) $beta$--$alpha$ }.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2026","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47242703","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}
In this note, we develop Fourier approximation methods for the solutions of first-order nonlocal mean-field games (MFG) systems. Using Fourier expansion techniques, we approximate a given MFG system by a simpler one that is equivalent to a convex optimization problem over a finite-dimensional subspace of continuous curves. Furthermore, we perform a time-discretization for this optimization problem and arrive at a finite-dimensional saddle point problem. Finally, we solve this saddle-point problem by a variant of a primal dual hybrid gradient method.
{"title":"Fourier approximation methods for first-order nonlocal mean-field games","authors":"L. Nurbekyan, João Saúde","doi":"10.4171/pm/2023","DOIUrl":"https://doi.org/10.4171/pm/2023","url":null,"abstract":"In this note, we develop Fourier approximation methods for the solutions of first-order nonlocal mean-field games (MFG) systems. Using Fourier expansion techniques, we approximate a given MFG system by a simpler one that is equivalent to a convex optimization problem over a finite-dimensional subspace of continuous curves. Furthermore, we perform a time-discretization for this optimization problem and arrive at a finite-dimensional saddle point problem. Finally, we solve this saddle-point problem by a variant of a primal dual hybrid gradient method.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2023","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47403921","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 study the homogenization of obstacle problems in Orlicz-Sobolev spaces for a wide class of monotone operators (possibly degenerate or singular) of the $p(cdot)$-Laplacian type. Our approach is based on the Lewy-Stampacchia inequalities, which then give access to a compactness argument. We also prove the convergence of the coincidence sets under non-degeneracy conditions.
{"title":"Homogenization of obstacle problems in Orlicz–Sobolev spaces","authors":"Diego Marcon, J. Rodrigues, R. Teymurazyan","doi":"10.4171/pm/2019","DOIUrl":"https://doi.org/10.4171/pm/2019","url":null,"abstract":"We study the homogenization of obstacle problems in Orlicz-Sobolev spaces for a wide class of monotone operators (possibly degenerate or singular) of the $p(cdot)$-Laplacian type. Our approach is based on the Lewy-Stampacchia inequalities, which then give access to a compactness argument. We also prove the convergence of the coincidence sets under non-degeneracy conditions.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43830125","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}
Let $T$ be an algebraic torus over an algebraically closed field, let $X$ be a smooth closed subvariety of a $T$-toric variety such that $U = X cap T$ is not empty, and let $mathscr{L}(X)$ be the arc scheme of $X$. We define a tropicalization map on $mathscr{L}(X) setminus mathscr{L}(X setminus U)$, the set of arcs of $X$ that do not factor through $X setminus U$. We show that each fiber of this tropicalization map is a constructible subset of $mathscr{L}(X)$ and therefore has a motivic volume. We prove that if $U$ has a compactification with simple normal crossing boundary, then the generating function for these motivic volumes is rational, and we express this rational function in terms of certain lattice maps constructed in Hacking, Keel, and Tevelev's theory of geometric tropicalization. We explain how this result, in particular, gives a formula for Denef and Loeser's motivic zeta function of a polynomial. To further understand this formula, we also determine precisely which lattice maps arise in the construction of geometric tropicalization.
设$T$是代数闭域上的代数环面，设$X$是$T$的光滑闭子簇，使得$U = X cap T$不为空，设$mathscr{L}(X)$是$X$的弧格式。我们在$mathscr{L}(X) setminus mathscr{L}(X setminus U)$上定义了一个热带化映射，即$X$中不经过$X setminus U$因式分解的弧的集合。我们证明了这个热带化图的每个纤维都是$mathscr{L}(X)$的一个可构造子集，因此具有一个动机体积。我们证明了如果$U$具有简单法向交叉边界紧化，那么这些动力体积的生成函数是有理的，并且我们用Hacking, Keel和Tevelev的几何热带化理论中构造的晶格映射来表示这个有理函数。我们特别解释了这个结果是如何给出Denef和Loeser的多项式的动机zeta函数的公式的。为了进一步理解这个公式，我们还精确地确定几何热带化构造中出现的点阵图。
{"title":"Motivic volumes of fibers of tropicalization","authors":"Jeremy Usatine","doi":"10.4171/pm/2045","DOIUrl":"https://doi.org/10.4171/pm/2045","url":null,"abstract":"Let $T$ be an algebraic torus over an algebraically closed field, let $X$ be a smooth closed subvariety of a $T$-toric variety such that $U = X cap T$ is not empty, and let $mathscr{L}(X)$ be the arc scheme of $X$. We define a tropicalization map on $mathscr{L}(X) setminus mathscr{L}(X setminus U)$, the set of arcs of $X$ that do not factor through $X setminus U$. We show that each fiber of this tropicalization map is a constructible subset of $mathscr{L}(X)$ and therefore has a motivic volume. We prove that if $U$ has a compactification with simple normal crossing boundary, then the generating function for these motivic volumes is rational, and we express this rational function in terms of certain lattice maps constructed in Hacking, Keel, and Tevelev's theory of geometric tropicalization. We explain how this result, in particular, gives a formula for Denef and Loeser's motivic zeta function of a polynomial. To further understand this formula, we also determine precisely which lattice maps arise in the construction of geometric tropicalization.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42685088","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 derive optimal a priori and a posteriori error estimates for Nitsche's method applied to unilateral contact problems. Our analysis is based on the interpretation of Nitsche's method as a stabilised finite element method for the mixed Lagrange multiplier formulation of the contact problem wherein the Lagrange multiplier has been eliminated elementwise. To simplify the presentation, we focus on the scalar Signorini problem and outline only the proofs of the main results since most of the auxiliary results can be traced to our previous works on the numerical approximation of variational inequalities. We end the paper by presenting results of our numerical computations which corroborate the efficiency and reliability of the a posteriori estimators.
{"title":"Nitsche’s method for unilateral contact problems","authors":"T. Gustafsson, R. Stenberg, J. Videman","doi":"10.4171/pm/2016","DOIUrl":"https://doi.org/10.4171/pm/2016","url":null,"abstract":"We derive optimal a priori and a posteriori error estimates for Nitsche's method applied to unilateral contact problems. Our analysis is based on the interpretation of Nitsche's method as a stabilised finite element method for the mixed Lagrange multiplier formulation of the contact problem wherein the Lagrange multiplier has been eliminated elementwise. To simplify the presentation, we focus on the scalar Signorini problem and outline only the proofs of the main results since most of the auxiliary results can be traced to our previous works on the numerical approximation of variational inequalities. We end the paper by presenting results of our numerical computations which corroborate the efficiency and reliability of the a posteriori estimators.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2018-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.4171/pm/2016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45670296","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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