Pub Date : 2025-11-04DOI: 10.1016/j.na.2025.114002
Stefano Marò , Rafael Ortega
We consider a class of periodic solutions of second order difference equations with symplectic structure. We obtain an explicit condition for their stability in terms of the 4-jet of the generating function. This result is applied to the discrete Newton equation, the model of a bouncing ball and the Fermi–Ulam ping-pong model.
{"title":"Stability of some periodic configurations of discrete Lagrangian equations","authors":"Stefano Marò , Rafael Ortega","doi":"10.1016/j.na.2025.114002","DOIUrl":"10.1016/j.na.2025.114002","url":null,"abstract":"<div><div>We consider a class of periodic solutions of second order difference equations with symplectic structure. We obtain an explicit condition for their stability in terms of the 4-jet of the generating function. This result is applied to the discrete Newton equation, the model of a bouncing ball and the Fermi–Ulam ping-pong model.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114002"},"PeriodicalIF":1.3,"publicationDate":"2025-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.na.2025.113982
Paolo Marcellini , Antonella Nastasi , Cintia Pacchiano Camacho
We propose some general growth conditions on the function , including the so-called natural growth, or polynomial, or growth conditions, or even exponential growth, in order to obtain that any local minimizer of the energy integral is locally Lipschitz continuous in . In fact this is the fundamental step for further regularity: the local boundedness of the gradient of any Lipschitz continuous local minimizer a-posteriori makes irrelevant the behavior of the integrand as ; i.e., the general growth conditions a posteriori are reduced to a standard growth, with the possibility to apply the classical regularity theory. In other words, we reduce some classes of non-uniform elliptic variational problems to a context of uniform ellipticity.
{"title":"Unified a-priori estimates for minimizers under p,q−growth and exponential growth","authors":"Paolo Marcellini , Antonella Nastasi , Cintia Pacchiano Camacho","doi":"10.1016/j.na.2025.113982","DOIUrl":"10.1016/j.na.2025.113982","url":null,"abstract":"<div><div>We propose some <em>general growth conditions</em> on the function <span><math><mrow><mi>f</mi><mo>=</mo><mi>f</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>ξ</mi></mrow></mfenced></mrow></math></span>, including the so-called <em>natural growth</em>, or <em>polynomial</em>, or <span><math><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>−</mo></mrow></math></span><em>growth conditions</em>, or even <em>exponential growth</em>, in order to obtain that any local minimizer of the energy integral <span><math><mrow><mspace></mspace><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>f</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>D</mi><mi>u</mi></mrow></mfenced><mi>d</mi><mi>x</mi><mspace></mspace></mrow></math></span> is <em>locally Lipschitz continuous</em> in <span><math><mi>Ω</mi></math></span>. In fact this is the fundamental step for further regularity: the <em>local boundedness of the gradient</em> of any Lipschitz continuous local minimizer <em>a-posteriori</em> makes irrelevant the behavior of the integrand <span><math><mrow><mi>f</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mi>ξ</mi></mrow></mfenced></mrow></math></span> as <span><math><mrow><mfenced><mrow><mi>ξ</mi></mrow></mfenced><mo>→</mo><mo>+</mo><mi>∞</mi></mrow></math></span>; i.e., the <em>general growth conditions</em> a posteriori are reduced to a standard growth, with the possibility to apply the classical regularity theory. In other words, we reduce some classes of <em>non-uniform</em> elliptic variational problems to a context of <em>uniform</em> ellipticity.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113982"},"PeriodicalIF":1.3,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145419857","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01DOI: 10.1016/j.na.2025.113986
Umberto Guarnotta , Patrick Winkert
In this paper we study quasilinear elliptic Kirchhoff equations driven by a non-homogeneous operator with unbalanced growth and right-hand sides that consist of sub-linear, possibly singular, and super-linear reaction terms. Under very general assumptions we prove the existence of at least two solutions for such problems by using the fibering method along with an appropriate splitting of the associated Nehari manifold. In contrast to other works our treatment is very general, with much easier and shorter proofs as it was done in the literature before. Furthermore, the results presented in this paper cover a large class of second-order differential operators like the -Laplacian, the -Laplacian, the double phase operator, and the logarithmic double phase operator.
{"title":"Degenerate singular Kirchhoff problems in Musielak–Orlicz spaces","authors":"Umberto Guarnotta , Patrick Winkert","doi":"10.1016/j.na.2025.113986","DOIUrl":"10.1016/j.na.2025.113986","url":null,"abstract":"<div><div>In this paper we study quasilinear elliptic Kirchhoff equations driven by a non-homogeneous operator with unbalanced growth and right-hand sides that consist of sub-linear, possibly singular, and super-linear reaction terms. Under very general assumptions we prove the existence of at least two solutions for such problems by using the fibering method along with an appropriate splitting of the associated Nehari manifold. In contrast to other works our treatment is very general, with much easier and shorter proofs as it was done in the literature before. Furthermore, the results presented in this paper cover a large class of second-order differential operators like the <span><math><mi>p</mi></math></span>-Laplacian, the <span><math><mrow><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></math></span>-Laplacian, the double phase operator, and the logarithmic double phase operator.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113986"},"PeriodicalIF":1.3,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145419858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-29DOI: 10.1016/j.na.2025.113999
Jiechen Qiang , Zhongwei Tang , Yichen Zhang
In this paper we first study the non-degeneracy of solutions to the critical CR-Yamabe type problem on the Heisenberg group. And as an application of this non-degeneracy, we study the existence of concentrating solutions to the slightly sub-critical problem involving the sub-Laplacian on a bounded domain of Heisenberg group. We construct sign-changing solutions as the parameter is sufficiently small under certain assumptions. Moreover, the solutions have precisely two nodal domains.
{"title":"On the non-degeneracy and existence of sign-changing solutions to elliptic problem on the Heisenberg group","authors":"Jiechen Qiang , Zhongwei Tang , Yichen Zhang","doi":"10.1016/j.na.2025.113999","DOIUrl":"10.1016/j.na.2025.113999","url":null,"abstract":"<div><div>In this paper we first study the non-degeneracy of solutions to the critical CR-Yamabe type problem on the Heisenberg group. And as an application of this non-degeneracy, we study the existence of concentrating solutions to the slightly sub-critical problem involving the sub-Laplacian on a bounded domain of Heisenberg group. We construct sign-changing solutions as the parameter is sufficiently small under certain assumptions. Moreover, the solutions have precisely two nodal domains.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113999"},"PeriodicalIF":1.3,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145419856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-29DOI: 10.1016/j.na.2025.113985
Hirokazu Saito , Yoshihiro Shibata
This paper is concerned with the global solvability for the Navier–Stokes equations describing viscous free surface flows of infinite depth in three and higher dimensions. In the finite-depth case, Poincaré inequalities play a crucial role to handle lower order terms in previous works such as Beale (1984), Guo and Tice (2013) in -based Sobolev spaces. Our situation of the present paper, however, cannot use them due to lack of boundedness in the vertical direction. To overcome this difficultly, we employ time decay estimates of a -analytic semigroup generated by the Stokes operator with a free boundary condition. More precisely, we first prove a time weighted estimate of solutions to a linearized system of the Navier–Stokes equations by using the time decay estimates as above and maximal regularity estimates in an -in-time and -in-space setting with suitable , . The time weighted estimate then enables us to show the global solvability of the Navier–Stokes equations for small initial data by the contraction mapping principle. Although this approach is based on our previous paper studying the three-dimensional case, we introduce a new time weighted estimate to deal with higher dimensions and simplify the proof of the three-dimensional case. Furthermore, we establish a new estimate of Duhamel’s integral based on Shibata (2022) in our linear theory, which is of independent interest.
{"title":"Global solvability for viscous free surface flows of infinite depth in three and higher dimensions","authors":"Hirokazu Saito , Yoshihiro Shibata","doi":"10.1016/j.na.2025.113985","DOIUrl":"10.1016/j.na.2025.113985","url":null,"abstract":"<div><div>This paper is concerned with the global solvability for the Navier–Stokes equations describing viscous free surface flows of infinite depth in three and higher dimensions. In the finite-depth case, Poincaré inequalities play a crucial role to handle lower order terms in previous works such as Beale (1984), Guo and Tice (2013) in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-based Sobolev spaces. Our situation of the present paper, however, cannot use them due to lack of boundedness in the vertical direction. To overcome this difficultly, we employ time decay estimates of a <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-analytic semigroup generated by the Stokes operator with a free boundary condition. More precisely, we first prove a time weighted estimate of solutions to a linearized system of the Navier–Stokes equations by using the time decay estimates as above and maximal regularity estimates in an <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-in-time and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-in-space setting with suitable <span><math><mi>p</mi></math></span>, <span><math><mi>q</mi></math></span>. The time weighted estimate then enables us to show the global solvability of the Navier–Stokes equations for small initial data by the contraction mapping principle. Although this approach is based on our previous paper studying the three-dimensional case, we introduce a new time weighted estimate to deal with higher dimensions and simplify the proof of the three-dimensional case. Furthermore, we establish a new estimate of Duhamel’s integral based on Shibata (2022) in our linear theory, which is of independent interest.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113985"},"PeriodicalIF":1.3,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145419859","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-27DOI: 10.1016/j.na.2025.114001
Yinxi Chen , Xingyu Liu
Our aim in this paper is to investigate the long-time behaviors at infinity of solutions to elliptic equations associated with Lévy operators. Utilizing the regularization method, we establish Schauder-type estimates near the flat boundary. Furthermore, we derive a Liouville-type result for Lévy operators, contributing to the broader theoretical framework of degenerate Lévy Ornstein–Uhlenbeck operators.
{"title":"Schauder type estimates and long-time behavior for elliptic equations associated with Lévy operators","authors":"Yinxi Chen , Xingyu Liu","doi":"10.1016/j.na.2025.114001","DOIUrl":"10.1016/j.na.2025.114001","url":null,"abstract":"<div><div>Our aim in this paper is to investigate the long-time behaviors at infinity of solutions to elliptic equations associated with Lévy operators. Utilizing the regularization method, we establish Schauder-type estimates near the flat boundary. Furthermore, we derive a Liouville-type result for Lévy operators, contributing to the broader theoretical framework of degenerate Lévy Ornstein–Uhlenbeck operators.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114001"},"PeriodicalIF":1.3,"publicationDate":"2025-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145419855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-24DOI: 10.1016/j.na.2025.113983
Nicolas Beuvin, Alberto Farina
We prove new one-dimensional symmetry results for non-negative solutions, possibly unbounded, to the semilinear equation in the upper half-space . Some Liouville-type theorems are also proven in the case of differential inequalities in , even without imposing any boundary condition.
Although subject to dimensional restrictions, our results apply to a broad family of functions . In particular, they apply to all non-negative that behaves at least linearly at infinity.
{"title":"One-dimensional symmetry results for semilinear equations and inequalities on half-spaces","authors":"Nicolas Beuvin, Alberto Farina","doi":"10.1016/j.na.2025.113983","DOIUrl":"10.1016/j.na.2025.113983","url":null,"abstract":"<div><div>We prove new one-dimensional symmetry results for non-negative solutions, possibly unbounded, to the semilinear equation <span><math><mrow><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></math></span> in the upper half-space <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>N</mi></mrow></msubsup></math></span>. Some Liouville-type theorems are also proven in the case of differential inequalities in <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>N</mi></mrow></msubsup></math></span>, even without imposing any boundary condition.</div><div>Although subject to dimensional restrictions, our results apply to a broad family of functions <span><math><mi>f</mi></math></span>. In particular, they apply to all non-negative <span><math><mi>f</mi></math></span> that behaves at least linearly at infinity.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113983"},"PeriodicalIF":1.3,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145340633","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-24DOI: 10.1016/j.na.2025.113984
Wojciech Górny , Michał Łasica , Alexandros Matsoukas
We consider a class of integral functionals with Musielak–Orlicz type variable growth, possibly linear in some regions of the domain. This includes power-type integrands with as well as double-phase integrands with . The main goal of this paper is to identify the -subdifferential of the functional, including a local characterisation in terms of a variant of the Anzellotti product defined through the Young’s inequality. As an application, we obtain the Euler–Lagrange equation for the variant of Rudin–Osher–Fatemi image denoising problem with variable growth regularising term. Moreover, we provide a characterisation of the -gradient flow of variable-growth total variation in terms of a parabolic PDE.
{"title":"Euler–Lagrange equations for variable-growth total variation","authors":"Wojciech Górny , Michał Łasica , Alexandros Matsoukas","doi":"10.1016/j.na.2025.113984","DOIUrl":"10.1016/j.na.2025.113984","url":null,"abstract":"<div><div>We consider a class of integral functionals with Musielak–Orlicz type variable growth, possibly linear in some regions of the domain. This includes <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span> power-type integrands with <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≥</mo><mn>1</mn></mrow></math></span> as well as double-phase <span><math><mrow><mi>p</mi><mspace></mspace><mo>−</mo><mspace></mspace><mi>q</mi></mrow></math></span> integrands with <span><math><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow></math></span>. The main goal of this paper is to identify the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-subdifferential of the functional, including a local characterisation in terms of a variant of the Anzellotti product defined through the Young’s inequality. As an application, we obtain the Euler–Lagrange equation for the variant of Rudin–Osher–Fatemi image denoising problem with variable growth regularising term. Moreover, we provide a characterisation of the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-gradient flow of variable-growth total variation in terms of a parabolic PDE.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113984"},"PeriodicalIF":1.3,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-24DOI: 10.1016/j.na.2025.113968
Ruhua Zhang , Guanggui Chen
In this paper, we establish the Hörmander type multiplier theorem for Fourier multipliers on Hardy spaces for , with regularity condition formulated in terms of modulation spaces where . We further investigate the boundedness of Fourier multipliers on Lebesgue spaces for through the interpolation. The conditions proposed in this paper not only improve those established by previous researchers but also refine the corresponding conclusions. Additionally, we introduce a novel multiplier theorem that incorporates the regularity condition formulated in terms of Wiener amalgam spaces . Here the multiplier theorem may be of methodology to further studies of Fourier multipliers.
{"title":"On the boundedness of Fourier multipliers in terms of modulation spaces regularity","authors":"Ruhua Zhang , Guanggui Chen","doi":"10.1016/j.na.2025.113968","DOIUrl":"10.1016/j.na.2025.113968","url":null,"abstract":"<div><div>In this paper, we establish the Hörmander type multiplier theorem for Fourier multipliers on Hardy spaces <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup><mfenced><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfenced></mrow></math></span> for <span><math><mrow><mn>0</mn><mo><</mo><mi>p</mi><mo>≤</mo><mn>1</mn></mrow></math></span>, with regularity condition formulated in terms of modulation spaces <span><math><mrow><msubsup><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mfenced><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfenced></mrow></math></span> where <span><math><mrow><mn>1</mn><mo>≤</mo><mi>r</mi><mo>,</mo><mi>q</mi><mo>≤</mo><mi>∞</mi><mo>,</mo><mi>s</mi><mo>></mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>p</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mi>q</mi></mrow></mfrac></mrow></math></span>. We further investigate the boundedness of Fourier multipliers on Lebesgue spaces <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mfenced><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfenced></mrow></math></span> for <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></math></span> through the interpolation. The conditions proposed in this paper not only improve those established by previous researchers but also refine the corresponding conclusions. Additionally, we introduce a novel multiplier theorem that incorporates the regularity condition formulated in terms of Wiener amalgam spaces <span><math><mrow><msubsup><mrow><mi>W</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>r</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mfenced><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfenced></mrow></math></span>. Here the multiplier theorem may be of methodology to further studies of Fourier multipliers.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"263 ","pages":"Article 113968"},"PeriodicalIF":1.3,"publicationDate":"2025-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145364204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-10-23DOI: 10.1016/j.na.2025.113987
Francesca Da Lio, Tristan Rivière, Dominik Schlagenhauf
In this paper we consider sequences of -harmonic maps, , from a closed Riemann surface into the -dimensional sphere with uniform bounded energy. These are critical points of the energy Our two main results are an improved pointwise estimate of the gradient in the neck regions around blow up points and the proof that the necks are asymptotically not contributing to the negativity of the second variation of the energy This allows us, in the spirit of the paper of the first and second authors in collaboration with Gianocca et al. (2022) , to show the upper semicontinuity of the Morse index plus nullity for sequences of -harmonic maps into a sphere.
{"title":"Morse index stability for Sacks–Uhlenbeck approximations for harmonic maps into a sphere","authors":"Francesca Da Lio, Tristan Rivière, Dominik Schlagenhauf","doi":"10.1016/j.na.2025.113987","DOIUrl":"10.1016/j.na.2025.113987","url":null,"abstract":"<div><div>In this paper we consider sequences of <span><math><mi>p</mi></math></span>-harmonic maps, <span><math><mrow><mi>p</mi><mo>></mo><mn>2</mn></mrow></math></span>, from a closed Riemann surface <span><math><mi>Σ</mi></math></span> into the <span><math><mi>n</mi></math></span>-dimensional sphere <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with uniform bounded energy. These are critical points of the energy <span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>≔</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>Σ</mi></mrow></msub><msup><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mrow><mrow><mo>|</mo><mo>∇</mo><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow><mrow><mi>p</mi><mo>/</mo><mn>2</mn></mrow></msup><mspace></mspace><mi>d</mi><mi>v</mi><mi>o</mi><msub><mrow><mi>l</mi></mrow><mrow><mi>Σ</mi></mrow></msub><mo>.</mo></mrow></math></span> Our two main results are an improved pointwise estimate of the gradient in the neck regions around blow up points and the proof that the necks are asymptotically not contributing to the negativity of the second variation of the energy <span><math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>.</mo></mrow></math></span> This allows us, in the spirit of the paper of the first and second authors in collaboration with Gianocca et al. (2022) , to show the upper semicontinuity of the Morse index plus nullity for sequences of <span><math><mi>p</mi></math></span>-harmonic maps into a sphere.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 113987"},"PeriodicalIF":1.3,"publicationDate":"2025-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145340632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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