{"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
{"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}
{"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
{"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}
{"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}
{"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}
{"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}