Pub Date : 2026-04-01Epub Date: 2025-11-19DOI: 10.1016/j.na.2025.114007
Jin Tan
We show the global-in-time existence and uniqueness of solutions to the 3D incompressible Hall-magnetohydrodynamic (Hall-MHD) system with small initial data in critical Sobolev spaces. Our result works for general physical parameters, thus gives a full answer to the problem proposed by Chae and Lee in the Remark 2 of Chae and Lee (2014). Moreover, considering the so-called 2D flows for the Hall-MHD system (that is 3D flows independent of the vertical variable), we show that under the sole assumption that the initial magnetic field is small in the critical Sobolev space leads to a global unique solvability statement. Comparing with the classical MHD system, the new difficulties of proving such results come from the additional Hall term, which endows the magnetic equation with a quasi-linear character.
{"title":"Global Fujita-Kato solutions for the incompressible Hall-MHD system","authors":"Jin Tan","doi":"10.1016/j.na.2025.114007","DOIUrl":"10.1016/j.na.2025.114007","url":null,"abstract":"<div><div>We show the global-in-time existence and uniqueness of solutions to the 3D incompressible Hall-magnetohydrodynamic (Hall-MHD) system with small initial data in critical Sobolev spaces. Our result works for general physical parameters, thus gives a full answer to the problem proposed by Chae and Lee in the Remark 2 of Chae and Lee (2014). Moreover, considering the so-called 2<span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>D flows for the Hall-MHD system (that is 3D flows independent of the vertical variable), we show that under the sole assumption that the initial magnetic field is small in the critical Sobolev space leads to a global unique solvability statement. Comparing with the classical MHD system, the new difficulties of proving such results come from the additional Hall term, which endows the magnetic equation with a quasi-linear character.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"265 ","pages":"Article 114007"},"PeriodicalIF":1.3,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145537289","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 : 2026-03-01Epub Date: 2025-11-05DOI: 10.1016/j.na.2025.114000
V. Angulo-Castillo , L.C.F. Ferreira , E.J. Villamizar-Roa
This paper addresses the -dimensional elliptic–parabolic Keller–Segel system by employing a Fourier-based framework. This approach allows to consider a new class of -initial data for the associated Cauchy problem, which can be arbitrarily large in norm. More specifically, our main result establishes global-in-time well-posedness for initial data belonging to a space of tempered distributions whose Fourier transform is a Radon measure supported on sum-closed frequency sets at a distance from the origin. The construction of large solutions can be carried out by taking the distance sufficiently large, a feasible condition that can be handled in practice through suitable translations in Fourier variables. The initial class includes, in particular, data of non-decaying type as as well as periodic and almost periodic functions.
{"title":"Large global bounded solutions for the parabolic–elliptic Keller–Segel system via a framework of sum-closed frequency sets","authors":"V. Angulo-Castillo , L.C.F. Ferreira , E.J. Villamizar-Roa","doi":"10.1016/j.na.2025.114000","DOIUrl":"10.1016/j.na.2025.114000","url":null,"abstract":"<div><div>This paper addresses the <span><math><mi>n</mi></math></span>-dimensional elliptic–parabolic Keller–Segel system by employing a Fourier-based framework. This approach allows to consider a new class of <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span>-initial data for the associated Cauchy problem, which can be arbitrarily large in norm. More specifically, our main result establishes global-in-time well-posedness for initial data belonging to a space of tempered distributions whose Fourier transform is a Radon measure supported on sum-closed frequency sets at a distance <span><math><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow></math></span> from the origin. The construction of large solutions can be carried out by taking the distance <span><math><mi>δ</mi></math></span> sufficiently large, a feasible condition that can be handled in practice through suitable translations in Fourier variables. The initial class includes, in particular, data of non-decaying type as <span><math><mrow><mo>|</mo><mi>x</mi><mo>|</mo><mo>→</mo><mi>∞</mi><mo>,</mo></mrow></math></span> as well as periodic and almost periodic functions.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114000"},"PeriodicalIF":1.3,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467930","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 : 2026-03-01Epub 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":"2026-03-01","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 : 2026-03-01Epub 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":"2026-03-01","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 : 2026-03-01Epub 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":"2026-03-01","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 : 2026-03-01Epub Date: 2025-11-07DOI: 10.1016/j.na.2025.114005
Kush Kinra , Fernanda Cipriano
In this article, we consider a class of incompressible stochastic third-grade fluids (non-Newtonian fluids) equations on two- as well as three-dimensional Poincaré domains (which may be bounded or unbounded). Our aims are to study the well-posedness and asymptotic analysis for the solutions of the underlying system. Firstly, we prove that the underlying system defined on has a unique weak solution (in the analytic sense) under Dirichlet boundary condition and it also generates random dynamical system . Secondly, we consider the underlying system on bounded domains. Using the compact Sobolev embedding , we prove the existence of a unique random attractor for the underlying system on bounded domains with external forcing in . Thirdly, we consider the underlying system on unbounded Poincaré domains with external forcing in and show the existence of a unique random attractor. In order to obtain the existence of a unique random attractor on unbounded domains, due to the lack of compact Sobolev embedding , we use the uniform-tail estimates method which helps us to demonstrate the asymptotic compactness of . Note that due to the presence of several nonlinear terms in the underlying system, we are not able to use the energy equality method to obtain the asymptotic compactness of in unbounded domains, which makes the analysis of this work in unbounded domains more difficult and interesting. Finally, as a consequence of the existence of random attractors, we address the existence of invariant measures for underlying system. To the best of authors’ knowledge, this is the first work which consider a class of the 2D as well as 3D incompressible stochastic third-grade fluids equations and establish the existence of random attractor in bounded as well as unbounded domains. In addition, this is the first work which address the existence of invariant measures for underlying system on unbounded domains.
{"title":"Random dynamics and invariant measures for a class of non-Newtonian fluids of differential type on 2D and 3D Poincaré domains","authors":"Kush Kinra , Fernanda Cipriano","doi":"10.1016/j.na.2025.114005","DOIUrl":"10.1016/j.na.2025.114005","url":null,"abstract":"<div><div>In this article, we consider a class of incompressible stochastic third-grade fluids (non-Newtonian fluids) equations on two- as well as three-dimensional Poincaré domains <span><math><mi>O</mi></math></span> (which may be bounded or unbounded). Our aims are to study the well-posedness and asymptotic analysis for the solutions of the underlying system. Firstly, we prove that the underlying system defined on <span><math><mi>O</mi></math></span> has a unique weak solution (in the analytic sense) under Dirichlet boundary condition and it also generates random dynamical system <span><math><mi>Ψ</mi></math></span>. Secondly, we consider the underlying system on bounded domains. Using the compact Sobolev embedding <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>O</mi><mo>)</mo></mrow><mo>↪</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>O</mi><mo>)</mo></mrow></mrow></math></span>, we prove the existence of a unique random attractor for the underlying system on bounded domains with external forcing in <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>O</mi><mo>)</mo></mrow></mrow></math></span>. Thirdly, we consider the underlying system on unbounded Poincaré domains with external forcing in <span><math><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>O</mi><mo>)</mo></mrow></mrow></math></span> and show the existence of a unique random attractor. In order to obtain the existence of a unique random attractor on unbounded domains, due to the lack of compact Sobolev embedding <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>O</mi><mo>)</mo></mrow><mo>↪</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>O</mi><mo>)</mo></mrow></mrow></math></span>, we use the uniform-tail estimates method which helps us to demonstrate the asymptotic compactness of <span><math><mi>Ψ</mi></math></span>. Note that due to the presence of several nonlinear terms in the underlying system, we are not able to use the energy equality method to obtain the asymptotic compactness of <span><math><mi>Ψ</mi></math></span> in unbounded domains, which makes the analysis of this work in unbounded domains more difficult and interesting. Finally, as a consequence of the existence of random attractors, we address the existence of invariant measures for underlying system. To the best of authors’ knowledge, this is the first work which consider a class of the 2D as well as 3D incompressible stochastic third-grade fluids equations and establish the existence of random attractor in bounded as well as unbounded domains. In addition, this is the first work which address the existence of invariant measures for underlying system on unbounded domains.</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114005"},"PeriodicalIF":1.3,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145467936","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 : 2026-03-01Epub 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":"2026-03-01","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 : 2026-03-01Epub 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":"2026-03-01","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}
Pub Date : 2026-03-01Epub Date: 2025-11-14DOI: 10.1016/j.na.2025.114017
Junior da S. Bessa , João Vitor da Silva , Maria N.B. Frederico , Gleydson C. Ricarte
<div><div>In this manuscript, we establish global weighted Orlicz-Sobolev and variable exponent Morrey–Sobolev estimates for viscosity solutions to fully nonlinear parabolic equations subject to oblique boundary conditions on a portion of the boundary, within the following framework: <span><math><mfenced><mrow><mtable><mtr><mtd><mi>F</mi><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd><mtd><mo>=</mo></mtd><mtd><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mtd><mtd><mtext>in</mtext></mtd><mtd><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><mi>β</mi><mi>⋅</mi><mi>D</mi><mi>u</mi><mo>+</mo><mi>γ</mi><mi>u</mi></mtd><mtd><mo>=</mo></mtd><mtd><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mtd><mtd><mtext>on</mtext></mtd><mtd><msub><mrow><mi>S</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn></mtd><mtd><mtext>on</mtext></mtd><mtd><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span> where <span><math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> denotes the parabolic cylinder with spatial base <span><math><mi>Ω</mi></math></span> (a bounded domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>) and temporal height <span><math><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></math></span>, <span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>∂</mi><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span>. Additionally, <span><math><mi>f</mi></math></span> represents the source term of the parabolic equation, while the boundary data are given by <span><math><mi>β</mi></math></span>, <span><math><mi>γ</mi></math></span>, and <span><math><mi>g</mi></math></span>. Our first main result is a global weighted Orlicz–Sobolev estimate for the solution, obtained under asymptotic structural conditions on the differential operator and appropriate assumptions on the boundary data, assuming that the source term belongs to the corresponding weighted Orlicz space. Leveraging these estimates, we demonstrate several applications, including a density result
{"title":"Weighted Orlicz-Sobolev and variable exponent Morrey regularity for fully nonlinear parabolic PDEs with oblique boundary conditions and applications","authors":"Junior da S. Bessa , João Vitor da Silva , Maria N.B. Frederico , Gleydson C. Ricarte","doi":"10.1016/j.na.2025.114017","DOIUrl":"10.1016/j.na.2025.114017","url":null,"abstract":"<div><div>In this manuscript, we establish global weighted Orlicz-Sobolev and variable exponent Morrey–Sobolev estimates for viscosity solutions to fully nonlinear parabolic equations subject to oblique boundary conditions on a portion of the boundary, within the following framework: <span><math><mfenced><mrow><mtable><mtr><mtd><mi>F</mi><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>,</mo><mi>D</mi><mi>u</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>−</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd><mtd><mo>=</mo></mtd><mtd><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mtd><mtd><mtext>in</mtext></mtd><mtd><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><mi>β</mi><mi>⋅</mi><mi>D</mi><mi>u</mi><mo>+</mo><mi>γ</mi><mi>u</mi></mtd><mtd><mo>=</mo></mtd><mtd><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mtd><mtd><mtext>on</mtext></mtd><mtd><msub><mrow><mi>S</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mtd><mtd><mo>=</mo></mtd><mtd><mn>0</mn></mtd><mtd><mtext>on</mtext></mtd><mtd><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span> where <span><math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span> denotes the parabolic cylinder with spatial base <span><math><mi>Ω</mi></math></span> (a bounded domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow></math></span>) and temporal height <span><math><mrow><mi>T</mi><mo>></mo><mn>0</mn></mrow></math></span>, <span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mi>∂</mi><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo></mrow></mrow></math></span>, and <span><math><mrow><msub><mrow><mi>Ω</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mi>Ω</mi><mo>×</mo><mrow><mo>{</mo><mn>0</mn><mo>}</mo></mrow></mrow></math></span>. Additionally, <span><math><mi>f</mi></math></span> represents the source term of the parabolic equation, while the boundary data are given by <span><math><mi>β</mi></math></span>, <span><math><mi>γ</mi></math></span>, and <span><math><mi>g</mi></math></span>. Our first main result is a global weighted Orlicz–Sobolev estimate for the solution, obtained under asymptotic structural conditions on the differential operator and appropriate assumptions on the boundary data, assuming that the source term belongs to the corresponding weighted Orlicz space. Leveraging these estimates, we demonstrate several applications, including a density result ","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114017"},"PeriodicalIF":1.3,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520546","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 : 2026-03-01Epub Date: 2025-11-12DOI: 10.1016/j.na.2025.114006
Alessandra De Luca
In the present paper, which aims at representing an improvement of De Luca and Felli (2021), we prove the validity of the strong unique continuation property for solutions to some second order elliptic equations from the edge of a crack via a description of their local behaviour. In particular we relax the star-shapedness condition on the complement of the crack considered in De Luca and Felli (2021) by applying a suitable diffeomorphism which straightens the boundary of the crack before performing an approximation of the fractured domain needed to derive a monotonicity formula
{"title":"A note on unique continuation from the edge of a crack with no star-shapedness condition","authors":"Alessandra De Luca","doi":"10.1016/j.na.2025.114006","DOIUrl":"10.1016/j.na.2025.114006","url":null,"abstract":"<div><div>In the present paper, which aims at representing an improvement of De Luca and Felli (2021), we prove the validity of the strong unique continuation property for solutions to some second order elliptic equations from the edge of a crack via a description of their local behaviour. In particular we relax the star-shapedness condition on the complement of the crack considered in De Luca and Felli (2021) by applying a suitable diffeomorphism which straightens the boundary of the crack before performing an approximation of the fractured domain needed to derive a monotonicity formula</div></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":"264 ","pages":"Article 114006"},"PeriodicalIF":1.3,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145520547","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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