This note complements the paper [19] by proving a scattering statement for solutions of nonlinear Klein-Gordon equations with an internal mode in 3d. We show that small solutions exhibit growth around a one-dimensional set in frequency space and become of order one in L∞ after a short transient time. The dynamics are driven by the feedback of the internal mode into the equation for the field (continuous spectral) component. The main part of the proof consists of showing suitable smallness for a “good” component of the radiation field. This is done in two steps: first, using the machinery developed in [19], we reduce the problem to bounding a certain quadratic normal form correction. Then we control this latter by establishing some refined estimates for certain bilinear operators with singular kernels.
{"title":"Internal mode-induced growth in $3$d nonlinear Klein–Gordon equations","authors":"Tristan L'eger, F. Pusateri","doi":"10.4171/rlm/986","DOIUrl":"https://doi.org/10.4171/rlm/986","url":null,"abstract":"This note complements the paper [19] by proving a scattering statement for solutions of nonlinear Klein-Gordon equations with an internal mode in 3d. We show that small solutions exhibit growth around a one-dimensional set in frequency space and become of order one in L∞ after a short transient time. The dynamics are driven by the feedback of the internal mode into the equation for the field (continuous spectral) component. The main part of the proof consists of showing suitable smallness for a “good” component of the radiation field. This is done in two steps: first, using the machinery developed in [19], we reduce the problem to bounding a certain quadratic normal form correction. Then we control this latter by establishing some refined estimates for certain bilinear operators with singular kernels.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42168264","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":"Closed selections of Vitali’s type","authors":"Jeremy Mirmina, D. Puglisi","doi":"10.4171/rlm/956","DOIUrl":"https://doi.org/10.4171/rlm/956","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49615536","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}
G. Bonanno, R. Livrea, Vicentiu D. Rădulescu, A. Ambrosetti
— The variational methods are adopted for establishing the existence of at least two nontrivial solutions for a Dirichlet problem driven by a non-homogeneous di¤erential operator of p-Laplacian type. A large class of nonlinear terms is considered, covering the concave-convex case. In particular, two positive solutions to the problem are obtained under a ðp 1Þ-superlinear growth at infinity, provided that a behaviour less than ðp 1Þ-linear of the nonlinear term in a suitable set is requested.
{"title":"Non-homogeneous Dirichlet problems with concave-convex reaction","authors":"G. Bonanno, R. Livrea, Vicentiu D. Rădulescu, A. Ambrosetti","doi":"10.4171/rlm/959","DOIUrl":"https://doi.org/10.4171/rlm/959","url":null,"abstract":"— The variational methods are adopted for establishing the existence of at least two nontrivial solutions for a Dirichlet problem driven by a non-homogeneous di¤erential operator of p-Laplacian type. A large class of nonlinear terms is considered, covering the concave-convex case. In particular, two positive solutions to the problem are obtained under a ðp 1Þ-superlinear growth at infinity, provided that a behaviour less than ðp 1Þ-linear of the nonlinear term in a suitable set is requested.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47344783","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":"Carleson estimates for the singular parabolic $p$-Laplacian in time-dependent domains","authors":"U. Gianazza","doi":"10.4171/rlm/953","DOIUrl":"https://doi.org/10.4171/rlm/953","url":null,"abstract":"","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46277716","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 paper, we consider an anisotropic Robin problem driven by the $p(x)$-Laplacian and a superlinear reaction. Applying variational tools along with truncation and comparison techniques as well as critical groups, we prove that the problem has at least five nontrivial smooth solutions to be ordered and with sign information: two positive, two negative, and the fifth nodal.
{"title":"A multiplicity theorem for anisotropic Robin equations","authors":"Nikolaos S. Papageorgiou, Patrick Winkert","doi":"10.4171/rlm/961","DOIUrl":"https://doi.org/10.4171/rlm/961","url":null,"abstract":"In this paper, we consider an anisotropic Robin problem driven by the $p(x)$-Laplacian and a superlinear reaction. Applying variational tools along with truncation and comparison techniques as well as critical groups, we prove that the problem has at least five nontrivial smooth solutions to be ordered and with sign information: two positive, two negative, and the fifth nodal.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":"2012 343","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138523905","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}
H. Brezis, A. Seeger, Jean Van Schaftingen, Po-Lam Yung
. We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on R n , using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo-Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the L p norm of functions in L p ( R n ) .
． 利用合适差商的超水平集的大小，我们描述了R n上Sobolev函数和BV函数的一个最新的单参数表征族。这为Bourgain, Brezis和Mironescu的BBM公式提供了另一种观点，并在BV的情况下补充了Cohen, Dahmen, Daubechies和DeVore关于这类函数的小波系数大小的一些结果。然后给出了对Gagliardo-Nirenberg插值不等式的一个应用。我们还建立了有关的L p (R n)中函数的L p范数的单参数族公式。
{"title":"Sobolev spaces revisited","authors":"H. Brezis, A. Seeger, Jean Van Schaftingen, Po-Lam Yung","doi":"10.4171/RLM/976","DOIUrl":"https://doi.org/10.4171/RLM/976","url":null,"abstract":". We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on R n , using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo-Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the L p norm of functions in L p ( R n ) .","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48106316","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 that quasi-concave positive solutions to a class of quasi-linear elliptic equations driven by the $p$-Laplacian in convex bounded domains of the plane have only one critical point. As a consequence, we obtain strict concavity results for suitable transformations of these solutions.
{"title":"Uniqueness of the critical point for solutions of some p-Laplace equations in the plane","authors":"William Borrelli, S. Mosconi, M. Squassina","doi":"10.4171/rlm/997","DOIUrl":"https://doi.org/10.4171/rlm/997","url":null,"abstract":"We prove that quasi-concave positive solutions to a class of quasi-linear elliptic equations driven by the $p$-Laplacian in convex bounded domains of the plane have only one critical point. As a consequence, we obtain strict concavity results for suitable transformations of these solutions.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48174141","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 an analyticity result for the Dirichlet-Neumann operator under space periodic boundary conditions in any dimension in an unbounded domain with infinite depth. We derive an analytic bifurcation result of analytic Stokes waves –i.e. space periodic traveling solutions– of the water waves equations in deep water. MSC 2020: 76B15, 35B32, 35J05.
{"title":"On the analyticity of the Dirichlet–Neumann operator and Stokes waves","authors":"M. Berti, A. Maspero, Paolo Ventura","doi":"10.4171/rlm/983","DOIUrl":"https://doi.org/10.4171/rlm/983","url":null,"abstract":"We prove an analyticity result for the Dirichlet-Neumann operator under space periodic boundary conditions in any dimension in an unbounded domain with infinite depth. We derive an analytic bifurcation result of analytic Stokes waves –i.e. space periodic traveling solutions– of the water waves equations in deep water. MSC 2020: 76B15, 35B32, 35J05.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47735808","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}
on the bottom of a half (N + 1)-dimensional ball. The interest in such a type of equations and related regularity issues has developed starting from the pioneering paper [7], proving local Hölder continuity results and Harnack’s inequalities, and has grown significantly in recent years stimulated by the study of the fractional Laplacian in its realization as a Dirichlet-to-Neumann map [3]. In this context, among recent regularity results for problems of type (1)–(2), we mention [2] and [12] for Schauder and gradient estimates with A being the identity matrix and c ≡ 0. More general degenerate/singular equations of type (1), admitting a varying coefficient matrix A, are considered in [19, 20]. In [19], under suitable regularity assumptions on A and c, Hölder continuity and C-regularity are established for solutions to (1)–(2) in the case h ≡ g ≡ 0, which, up to a reflection through the hyperspace t = 0, corresponds to the study of solutions to the equation − div(|t|1−2sA∇U) + |t|1−2sc = 0 which are even with respect to the t-variable; Hölder continuity of solutions which are odd in t is instead investigated in [20]. In addition, in [19] C and C bounds are derived for some inhomogeneous Neumann boundary problems (i.e. for g 6≡ 0) in the case c ≡ 0. The goal of the present note is to derive Sobolev-type regularity results for solutions to (1)–(2). Under suitable assumptions on c, h, g, the presence of the singular/degenerate homogenous weight, involving only the (N +1)-th variable t, makes the solutions to have derivates with respect to the first N variables x1, x2, . . . , xN belonging to a weighted H -space (with the same weight t1−2s); concerning the regularity of the derivative with respect to t, we obtain instead that the weighted derivative t1−2s ∂U ∂t belongs to a H-space with the dual weight t2s−1, confirming what has already been observed in [19, Lemma 7.1] for even solutions of the reflected problem corresponding to (1)–(2) with h ≡ g ≡ 0.
在半（N+1）维球的底部。从证明局部Hölder连续性结果和Harnack不等式的开创性论文[7]开始，人们对这类方程和相关正则性问题的兴趣就得到了发展，近年来，由于对分数拉普拉斯算子实现为狄利克雷-诺依曼映射的研究，人们对它的兴趣显著增长[3]。在这种情况下，在最近关于（1）-（2）型问题的正则性结果中，我们提到了Schauder和梯度估计的[2]和[12]，其中A是单位矩阵，c≠0。在[19，20]中考虑了更一般的（1）型退化/奇异方程，允许变系数矩阵a。在[19]中，在关于A和c的适当正则性假设下，在H Select g Select 0的情况下，建立了（1）–（2）的解的Hölder连续性和c正则性，它直到通过超空间t=0的反射，对应于对方程−div（|t|1−2sAŞU）+|t|1−2sc=0的解的研究，这些解相对于t变量是偶数的；在[20]中研究了t中奇数解的Hölder连续性。此外，在[19]中，对于一些非齐次Neumann边界问题（即g6≠0），在C≠0的情况下，导出了C和C的界。本注释的目的是导出（1）-（2）解的Sobolev型正则性结果。在对c，h，g的适当假设下，仅涉及第（N+1）个变量t的奇异/退化齐次权的存在使得解具有关于前N个变量x1，x2，…的导数，xN属于加权的H-空间（具有相同的权重t1−2s）；关于导数相对于t的正则性，我们得到了加权导数t1−2sõUõt属于具有对偶权重t2s−1的H空间，证实了在[19，引理7.1]中已经观察到的关于对应于（1）-（2）的反射问题的偶数解的情况。
{"title":"Sobolev-type regularity and Pohozaev-type identities for some degenerate and singular problems","authors":"V. Felli, Giovanni Siclari","doi":"10.4171/rlm/980","DOIUrl":"https://doi.org/10.4171/rlm/980","url":null,"abstract":"on the bottom of a half (N + 1)-dimensional ball. The interest in such a type of equations and related regularity issues has developed starting from the pioneering paper [7], proving local Hölder continuity results and Harnack’s inequalities, and has grown significantly in recent years stimulated by the study of the fractional Laplacian in its realization as a Dirichlet-to-Neumann map [3]. In this context, among recent regularity results for problems of type (1)–(2), we mention [2] and [12] for Schauder and gradient estimates with A being the identity matrix and c ≡ 0. More general degenerate/singular equations of type (1), admitting a varying coefficient matrix A, are considered in [19, 20]. In [19], under suitable regularity assumptions on A and c, Hölder continuity and C-regularity are established for solutions to (1)–(2) in the case h ≡ g ≡ 0, which, up to a reflection through the hyperspace t = 0, corresponds to the study of solutions to the equation − div(|t|1−2sA∇U) + |t|1−2sc = 0 which are even with respect to the t-variable; Hölder continuity of solutions which are odd in t is instead investigated in [20]. In addition, in [19] C and C bounds are derived for some inhomogeneous Neumann boundary problems (i.e. for g 6≡ 0) in the case c ≡ 0. The goal of the present note is to derive Sobolev-type regularity results for solutions to (1)–(2). Under suitable assumptions on c, h, g, the presence of the singular/degenerate homogenous weight, involving only the (N +1)-th variable t, makes the solutions to have derivates with respect to the first N variables x1, x2, . . . , xN belonging to a weighted H -space (with the same weight t1−2s); concerning the regularity of the derivative with respect to t, we obtain instead that the weighted derivative t1−2s ∂U ∂t belongs to a H-space with the dual weight t2s−1, confirming what has already been observed in [19, Lemma 7.1] for even solutions of the reflected problem corresponding to (1)–(2) with h ≡ g ≡ 0.","PeriodicalId":54497,"journal":{"name":"Rendiconti Lincei-Matematica e Applicazioni","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48182046","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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